
Class 
Book 



Pla&A 




lb fold out. opposite Taga 614 



OUTLINES OF 
ASTRONOMY 



BY 



SIR JOHN F. W. HERSCHEL 



PART TWO 




NEW YORK 

P. F. COLLIER & SON 

M C M I I 






By Transfer 
Maritime Comm. 

SEP 3 1949 



U-J 



CONTENTS 



CHAPTER XI 

OF COMETS 

Great Number of Recorded Comets — The Number of those Unrecorded 
Probably much Greater — General Description of a Comet — Comets 
without Tails, or with more than One — Their Extreme Tenuity — 
Their Probable Structure — Motions Conformable to the Law of Grav- 
ity — Actual Dimensions of Comets — Periodical Return of Several — 
Halley's Comet — Other Ancient Comets Probably Periodic — Encke's 
Comet — Biela's — Faye's — Lexell's — De Yico's — Brorsen's — Peters's — 
Great Comet of 1843 — Its Probable Identity with Several Older Comets 
— Great Interest at Present Attached to Cometary Astronomy, and 
its Reasons — Remarks on Cometary Orbits in General — Great Comets, 
of 1858, 1861 and 1862 465 

PART II 

OF TEE LUNAR AND PLANETARY PERTURBATIONS 

CHAPTER XII 

Subject Propounded — Problem of Three Bodies — Superposition of Small 
Motions — Estimation of the Disturbing Force — Its Geometrical Rep- 
resentation — Numerical Estimation in Particular Cases — Resolution 
into Rectangular Components — Radial, Transversal and Orthogonal 
Disturbing Forces — Normal and Tangential — Their Characteristic Ef- 
fects — Effects of the Orthogonal Force — Motion of the Nodes — Con- 
ditions of their Advance and Recess — Cases of an Exterior Planet 
Disturbed by an Interior — The Reverse Case — In Every Case the 
Node of the Disturbed Orbit Recedes on the Plane of the Disturb- 
ing on an Average — Combined Effect of Many Such D.sturbances — 
Motion of the Moon's Nodes — Change of Inclination — Conditions of 
its Increase and Diminution — Average Effect in a Whole Revolution 
— Compensation in a Complete Revolution of the Nodes — Lagrange's 
Theorem of the Stability of the Inclinations of the Planetary Orbits 
— Change of Obliquity of the Ecliptic — Precession of the Equinoxes 
Explained — Nutation — Principle of Forced Vibrations . . . 513 

(Hi) 



IV CONTENTS 

CHAPTER XIII 

THEORY OP THE AXES, PERIHELIA, AND EXCENTRICITIES 

Variation of Elements in General — Distinction between Periodic and Sec- 
ular Variations — Geometrical Expression of Tangential and Normal 
Forces — Variation of the Major Axis Produced only by the Tangential 
Force — Lagrange's Theorem of the Conservation of the Mean Dis- 
tances and Periods — Theory of the Perihelia and Excentricities — 
Geometrical Representation of their Momentary Variations — Estima- 
tion of the Disturbing Forces in Nearly Circular Orbits — Application 
to the Case of the Moon — Theory of the Lunar Apsides and Excen- 
tricity — Experimental Illustration — Application of the Foregoing Prin- 
ciples to the Planetary Theory — Compensation in Orbits very nearly 
Circular — Effects of Ellipticity — General Results — Lagrange's Theorem 
of the Stability of the Excentricities 554 

CHAPTER XIV 

Of the Inequalities Independent of the Excentricities — The Moon's Varia- 
tion and Parallactic Inequality — Analogous Planetary Inequalities — 
Three Cases of Planetary Perturbation Distinguished — Of Inequalities 
Dependent on the Excentricities — Long Inequality of Jupiter and 
Saturn — Law of Reciprocity Between the Periodical Variations of the 
Elements of both Planets — Long Inequality of the Earth and Venus — 
Variation of the Epoch — Inequalities Incident on the Epoch Affecting 
the Mean Motion — Interpretation of the Constant Pare of these In- 
equalities — Annual Equation of the Moon — Her Secular Acceleration 
— Lunar Inequalities Due to the Action of Venus — Effect of the Sphe- 
roidal Figure of the Earth and Other Planets on the Motions of their 
Satellites — Of the Tides — Masses of Disturbing Bodies Deducible from 
the Perturbations they Produce — Mass of the Moon, and of Jupiter's 
Satellites, how Ascertained — Perturbations of Uranus Resulting in 
the Discovery of Neptune — Determination of the Absolute Mass and 
Density of the Earth 603 



PART III 

OF SIDEREAL ASTRONOMY 

CHAPTER XV 

Of the Fixed Stars — Their Classification by Magnitudes — Photometric Scale 
of Magnitudes — Conventional or Vulgar Scale — Photometric Compari- 
son of Stars — Distribution of Stars over the Heavens — Of the Milky 
Way or Galaxy — Its Supposed Form that of a Flat Stratum Partially 



CONTENTS V 

Subdivided — Its Visible Course among the Constellations — Its Internal 
Structure— Its Apparently Indefinite Extent in Certain Directions — Of 
the Distance of the Fixed Stars — Their Annual Parallax — Parallactic 
Unit of Sidereal Distance — Effect of Parallax Analogous to that of 
Aberration — How Distinguished from it — Detection of Parallax by 
Meridional Observations — Henderson's Application to <* Centauri — 
By Differential Observations— Discoveries of Bessel and Struve — 
List of Scars in which Parallax has been Detected — Of the Real 
Magnitudes of the Stars — Comparison of their Lights with that of 
the Sun 692 

CHAPTER XVI 

Variable and Periodical Stars — List of those Already Known — Irregulari- 
ties in their Periods and Lustre when Brightest — Irregular and Tem- 
porary Stars — Ancient Chinese Records of Several — Missing Stars — 
Double Stars — Their Classification — Specimens of each Class — Binary 
Systems — Revolution round each other — Describe Elliptic Orbits under 
the Newtonian Law of Gravity — Elements of Orbits of Several — Actual 
Dimensions of their Orbits — Colored Double Stars — Phenomenon of 
Complementary Colors — Sanguine Stars — Proper Motion of the Stars 
—Partly Accounted for by a Real Motion of the Sun — Situation of 
the Solar Apex — Agreement of Southern and Northern Stars in Giving 
the Same Result — Principles on which the Investigation of the Solar 
Motion Depends — Absolute Velocity of the Sun's Motion — Supposed 
Revolution of the "Whole Sidereal System Round a Common Centre 
— Systematic Parallax and Aberration — Effect of the Motion of Light, 
in Altering the Apparent Period of a Binary Star . . . . *?33 

CHAPTER XVII 

OF CLUSTERS OF STARS AND NEBULAE 

Of Clustering Groups of Stars — Globular Clusters — Their Stability Dynami- 
cally Possible — List of the most Remarkable — Classification of Nebulas 
and Clusters — Their Distribution over the Heavens — Irregular Clusters 
— Resolvability of Nebulas — Theory of the Formation of Clusters by 
Nebulous Subsidence— Of Elliptic Nebula? — That of Andromeda — An- 
nular and Planetary Nebulas — Double Nebulas — Nebulous Stars — Con- 
nection of Nebulas with Double Stars — Insulated Nebulas of Forms 
not Wholly Irregular — Of Amorphous Nebulas — Their Law of Distri- 
bution Marks them as Outliers of the Galaxy — Nebulas and Nebulous 
Group of Orion — Of Argo — Of Sagittarius — Of Cygnus — The Magel- 
lanic Clouds — Singular Nebula in the Greater of Them — Variable 
Nebulas — The Zodiacal Light — Shooting Stars — Speculations on the 
Dynamical Origin of the Sun's Heat 118 



VI CONTENTS 

PART IV 

OF THE ACCOUNT OF TIME 

CHAPTER XYIII 

Natural Units of Time — Relation of the Sidereal to the Solar Day Affected 
by Precession — Incommensurability of the Day and Year — Its Incon- 
venience — How Obviated — The Julian Calendar — Irregularities at its 
first Introduction — Reformed by Augustus — Gregorian Reformation — 
Solar and Lunar Cycles — Indiction — Julian Period — Table of Chrono- 
logical Eras — Rules for Calculating the Days Elapsed between Given 
Dates — Equinoctial Time — Fixation of Ancient Dates by Eclipses — Notes 
and Additions 818 

List op Plates 888 



APPENDIX 

I. Lists of Northern and Southern Stars, with their approximate Magni- 
tudes, on the Vulgar and Photometric Scale ..... 891 
II. Synoptic Table of the Elements of the Planetary System . . . 893 

III. Synoptic Table of the Elements of the Orbits of the Satellites, so far 

as they are known ......... 900 

IV. Elements of Periodical Comets 903 

Table of Numbers in frequent use among Astronomers .... 907 

Index . . 909 



OUTLINES OF ASTRONOMY 



CHAPTEK XI 

OF COMETS 

Great Number of Recorded Comets — The Number of those Unrecorded 
Probably much Greater — General Description of a Comet — Comets 
without Tails, or with more than One — Their Extreme Tenuity — 
Their Probable Structure — Motions Conformable to the Law of Grav- 
ity — Actual Dimensions of Comets — Periodical Return of Several — 
Halley's Comet — Other Ancient Comets Probably Periodic — Encke's 
Comet — Biela ' s — Fay e ' s — Lexell ' s — De Yi co' s — Brorsen' s — Pe ter s ' s — 
Great Comet of 1843 — Its Probable Identity with Several Older Comets 
— Great Interest at Present Attached to Cometary Astronomy, and 
its Reasons — Remarks on Cometary Orbits in General — Great Comets, 
of 1858, 1861 and 1862 

(554.) The extraordinary aspect of comets, their rapid 
and seemingly irregular motions, the unexpected manner in 
which they often burst upon us, and the imposing magni- 
tudes which they occasionally assume, have in all ages 
rendered them objects of astonishment, not unmixed with 
superstitious dread to the uninstructed, and an enigma to 
those most conversant with the wonders of creation and the 
operations of natural causes. Even now, that we have 
ceased to regard their movements as irregular, or as gov- 
erned by other laws than those which retain the planets in 
their orbits, their intimate nature, and the offices they per- 
form in the economy of our system, are as much unknown 
as ever. No distinct and satisfactory account has yet been 

(465) 



4:66 OUTLINES OF ASTRONOMY 

rendered of those immensely voluminous appendages which 
they bear about with them, and which are known by the 
name of their tails (though improperly, since they often 
precede them in their motions), any more than of several 
other singularities which they present. 

(555.) The number of comets which have been astro- 
nomically observed, or of which notices have been recorded 
in history, is very great, amounting to several hundreds, 1 
and when we consider that in the earlier ages of astron- 
omy, and indeed in more recent times, before the inven- 
tion of the telescope, only large and conspicuous ones were 
noticed; and that, since due attention has been paid to the 
subject, scarcely a year has passed without the observation 
of one or two of these bodies, and that sometimes two and 
even three have appeared at once; it will be easily supposed 
that their actual number must be at least many thousands. 
Multitudes, indeed, must escape all observation, by reason 
of their paths traversing only that part of the heavens 
which is above the horizon in the daytime. Comets so 
circumstanced can only become visible by the rare coinci- 
dence of a total eclipse of the sun — a coincidence which 
happened, as related by Seneca, sixty-two years before 
Christ, when a large comet was actually observed very 
near the sun. Several, however, stand on record as hav- 
ing been bright enough to be seen with the naked eye in 



1 See catalogues in the Almagest of Riccioli ; Pingre's Cometographie ; De- 
lambre's Astron. vol. iii. ; Astronomische Abhandlungen, No. 1 (which contains 
the elements of all the orbits of comets which have been computed to the time 
of its publication, 1823); also a catalogue, by the Eev. T. J. Hussey. Loud, 
and Ed. Phil. Mag. vol. ii. No. 9, et seq. In a list cited by Lalande from tha 
1st vol. of the Tables de Berlin, 700 comets are enumerated. See also notices 
of the Astronomical Society and Astron. Nachr. passim. A great many of the 
more ancient comets are recorded in the Chinese Annals, and in some cases with 
sufficient precision to allow of the calculation of rudely approximate orbits from 
their motions so described. 



OUTLINES OF ASTRONOMY 467 

the daytime, even at noon and in bright sunshine. Such 
were tlie comets of 1402, 1532 and 1843, and that of 43 B.C. 
which appeared during the games celebrated by Augustus 
in honor of Venus shortly after the death of Caesar, and 
which the flattery of poets declared to be the soul of that 
hero taking its place among the divinities. 

(556.) That feelings of awe and astonishment should 
be excited by the sudden and unexpected appearance of 
a great comet, is no way surprising; being, in fact, accord- 
ing to the accounts we have of such events, one of the 
most imposing of all natural phenomena. Comets consist 
for the most part of a large and more or less splendid, but 
ill-defined nebulous mass of light, called the head, which 
is usually much brighter toward its centre, and offers the 
appearance of a vivid nucleus, like a star or planet. From 
the head, and in a direction opposite to that in which the sun 
is situated from the comet appear to diverge two streams of 
light, which grow broader and more diffused at a distance 
from the head, and which most commonly close in and 
unite at a little distance behind it, but sometimes continue 
distinct for a great part of their course; producing an effect 
like that of the trains left by some bright meteors, or like 
the diverging fire of a sky-rocket (only without sparks or 
perceptible motion). This is the tail. This magnificent ap- 
pendage attains occasionally an immense apparent length. 
Aristotle relates of the tail of the comet of 371 B.C., that 
it occupied a third of the hemisphere, or 60°; that of A.D. 
1618 is stated to have been attended by a train no less than 
104° in length. The comet of 1680, the most celebrated of 
modern times, and on many accounts the most remarkable 
of all, with a head not exceeding in brightness a star of the 
second magnitude, covered with its tail an extent of more 



468 OUTLINES OF ASTRONOMY 

than 70° of the heavens, or, as some accounts state, 90°; 
that of the comet of 1769 extended 97°, and that of the last 
great comet (1843) was estimated at about 65° when longest. 
The figure (fig. 2, Plate II.) is a representation of the comet 
of 1819 — by no means one of the most considerable, but 
which was, however, very conspicuous to the naked eye. 

(557.) The tail is, however, by no means an invariable 
appendage of comets. Many of the brightest • have been 
observed to have short and feeble tails, and a few great 
comets have been entirely without them. Those of 1585 
and 1763 offered no vestige of a tail; and Cassini describes 
the comets of 1665 and 1682 as being as round 2 and as well 
defined as Jupiter. On the other hand, instances are not 
wanting of comets furnished with many tails or streams of 
diverging light. That of 1744 had no less than six, spread 
out like an immense fan, extending to a distance of nearly 
30° in length. The small comet of 1823 had two, making 
an angle of about 160°, the brighter turned as usual from 
the sun, the fainter toward it, or nearly so. The tails of 
comets, too, are often somewhat curved, bending, in gen- 
eral, toward the region which the comet has left, as if 
moving somewhat more slowly, or as if resisted in their 
course. 

(558.) The smaller comets, such as are visible only in 
telescopes, or with difficulty by the naked eye, and which 
are by far the most numerous, offer very frequently no 
appearance of a tail, and appear only as round or some- 



2 This description, however, applies to the "disk" of the head of these 
comets as seen in a telescope. Cassini's expressions are, "aussi rond, aussi 
net, et aussi clair que Jupiter" (where it is to be observed that the latter 
epithet must by no means be translated bright). To understand this passage 
fully, the reader must refer to the description given further on, of the "disk" 
of Halley's comet, after its perihelion passage in 1835-6. 



OUTLINES OF ASTRONOMY 469 

what oval vaporous masses, more dense toward the centre, 
where, however, they appear to have no distinct nucleus, or 
anything which seems entitled to be considered as a solid 
body. This was shown in a very remarkable manner in 
the case of the comet discovered by Miss Mitchell in 1847, 
which on the 5th of October in that year passed centrally 
over a star of the fifth, magnitude: so centrally that with a 
magnifying power of 100 it was impossible to determine in 
which direction the extent of the nebulosity was greatest. 
The star's light seemed in no degree enfeebled; yet such 
a star would be completely obliterated by a moderate fog, 
extending only a few yards from the surface of the earth. 
And since it is an observed fact, that even those larger 
comets which have presented the appearance of a nucleus 
have yet exhibited no phases, though we cannot doubt that 
they shine by the reflected solar light, it follows that even 
these can only be regarded as great masses of thin vapor, 
susceptible of being penetrated through their whole sub- 
stance by the sunbeams, and reflecting them alike from 
their interior parts and from their surfaces. Nor will any 
one regard this explanation as forced, or feel disposed to 
resort to a phosphorescent quality in the comet itself, to 
account for the phenomena in question, when we consider 
(what will be hereafter shown) the enormous magnitude of 
the space thus illuminated, and the extremely small mass 
which there is ground to attribute to these bodies. It will 
then be evident that the most unsubstantial clouds which 
float in the highest regions of our atmosphere, and seem 
at sunset to be drenched in light, and to glow throughout 
their whole depth as if in actual ignition, without any 
shadow or dark side, must be looked upon as dense and 
massive bodies compared with the filmy and all but spirit- 



470 OUTLINES OF ASTRONOMY 

ual texture of a comet. Accordingly, whenever powerful 
telescopes have been turned on these bodies, they have not 
failed to dispel the illusion which attributes solidity to that 
more condensed part of the head, which appears to the 
naked eye as a nucleus; though it is true that in some 
a very minute stellar point has been seen, indicating the 
existence of something more substantial. 

(559.) It is in all probability to the feeble coercion of 
the elastic power of their gaseous parts, by the gravitation 
of so small a central mass, that we must attribute this 
extraordinary development of the atmospheres of comets. 
If the earth, retaining its present size, weie reduced, by 
any internal change (as by hollowing out its central parts) 
to one thousandth part of its actual mass, its coercive power 
over the atmosphere would be diminished in the same pro- 
portion, and in consequence the latter would expand to a 
thousand times its actual bulk; and indeed much more, 
owing to the still further diminution of gravity, by the 
recess of the upper parts from the centre. 3 An atmos- 
phere, however, free to expand equally in all directions, 
would envelope the nucleus spherically, so that it becomes 
necessary to admit the action of other causes to account for 
its enormous extension in the direction of the tail — a subject 
to which we shall presently take occasion to recur. 

(560.) That the luminous part of a comet is something 
in the nature of a smoke, fog or cloud, suspended in a 
transparent atmosphere, is evident from a fact which has 



3 Newton has calculated (Prine. III. p. 512) that a globe of air of ordinary- 
density at the earth's surface, of one inch in diameter, if reduced to the density 
due to the altitude above the surface of one radius of the earth, would occupy a 
sphere exceeding in radius the orbit of Saturn. The tail of a great comet, then, 
for aught we can tell, may consist of only a very few pounds or even ounces 
of matter. 



OUTLINES OF ASTRONOMY 471 

often noticed, viz. — that the portion of the tail where 
it comes closest to, and surrounds the head, is yet separated 
from it by an interval less luminous, as if sustained and 
kept off from contact by a transparent stratum, as we often 
one layer of clouds over another with a considerable 
clear space between. These, and most of the other facts 
observed in the history of comets, appear to indicate that 
the structure of a comet, as seen in section in the direction 
of its length, must be that of a hollow envelope, of a para- 
bolic form, inclosing near its vertex the nucleus and head, 
something as represented in the annexed figure. This would 




account for the apparent division of the tail into two principal 
lateral branches, the envelope being oblique to the line of 
sight at its borders, and therefore a greater depth of illumi- 
nated matter being there exposed to the eye. In all proba- 
bility, however, they admit great varieties of structure, and 
among them may very possibly be bodies of widely differ- 
ent physical constitution, and there is no doubt that one and 
the same comet at different epochs undergoes great changes, 
both in the disposition of its materials and in their physical 
state. 

(561.) We come now to speak of the motions of comets. 
These are apparently most irregular and capricious. Some- 
times they remain in sight only for a few days, at others for 
many months; some move with extreme slowness, others 
with extraordinary velocity; while not infrequently, the 
two extremes of apparent speed are exhibited by the same 



472 OUTLINES OF ASTRONOMY 

comet in different parts of its course. The comet of 1472 
described an arc of the heavens of 40° of a great circle 4 in a 
single day. Some pursue a direct, some a retrograde, and 
others a tortuous and very irregular course; nor do they 
confine themselves, like the planets, within any certain 
region of the heavens, but traverse indifferently every part. 
Their variations in apparent size, during the time they con- 
tinue visible, are no less remarkable than those of their 
velocity; sometimes they make their first appearance as 
faint and slow-moving objects, with little or no tail; but 
by degrees accelerate, enlarge, and throw out from them 
this appendage, which increases in length and brightness till 
(as always happens in such cases) they approach the sun, 
and are lost in his beams. After a time they again emerge, 
on the other side, receding from the sun with a velocity at 
first rapid, but gradually decaying. It is for the most part 
after thus passing the sun, that they shine forth in all their 
splendor, and that their tails acquire their greatest length 
and development; thus indicating plainly the action of the 
sun's rays as the exciting cause of that extraordinary emana- 
tion. As they continue to recede from the sun, their motion 
diminishes and the tail dies away, or is absorbed into the 
head, which itself grows continually feebler, and is at length 
altogether lost sight of, in by far the greater number of cases 
never to be seen more. 

(562.) Without the clew furnished by the theory of gravi- 
tation, the enigma of these seemingly irregular and capri- 
cious movements might have remained for ever unresolved. 
But Newton, having demonstrated the possibility of any 
conic section whatever being described about the sun, by a 

4 120° in extent in the former editions. But this was the arc described in 
longitude, and the comet at the time referred to had great north latitude. 



OUTLINES OF ASTRONOMY 478 

body revolving under the dominion of that law, immedi- 
ately perceived the applicability of the general proposition 
to the case of cometary orbits; and the great comet of 1680, 
one of the most remarkable on record, both for the immense 
length of its tail and for the excessive closeness of its ap- 
proach to the sun (within one-sixth of the diameter of that 
luminary), afforded him an excellent opportunity for the 
trial of his theory. The success of the attempt was com- 
plete. He ascertained that this comet described about the 
sun as its focus an elliptic orbit of so great an excentricity 
as to be indistinguishable from a parabola (which is the ex- 
treme, or limiting form of the ellipse when the axis becomes 
infinite), and that in this orbit the areas described about the 
sun were, as in the planetary ellipses, proportional to the 
times. The representation of the apparent motions of this 
comet by such an orbit, throughout its whole observed 
course, was found to be as satisfactory as those of the mo- 
tions of the planets in their nearly circular paths. From 
that time it became a received truth, that the motions of 
comets are regulated by the same general laws as those 
of the planets — the difference of the cases consisting only 
in the extravagant elongation of their ellipses, and in the 
absence of any limit to the inclinations of their planes to that 
of the ecliptic, or of any general coincidence in the direc- 
tion of their motions from west to east, rather than from 
east to west, like what is observed among the planets. 

(563.) It is a problem of pure geometry, from the gen- 
eral laws of elliptic or parabolic motion, to find the situa- 
tion and dimensions of the ellipse or parabola which shall 
represent the motion of any given comet. In general, three 
complete observations of its right ascension and declination, 
with the times at which they were made, suffice for the solu- 



474 OUTLINES OF ASTRONOMY 

tion of this problem (which is, however, by no means an 
easy one) and for the determination of the elements of the 
orbit. These consist, mutatis mutandis, of the same data as 
are required for the computation of the motion of a planet 
(that is to say, the longitude of the perihelion, that of the 
ascending node, the inclination to the ecliptic, the semi-axis, 
excentricity, and time of perihelion passage, as also whether 
the motion is direct or retrograde); and, once determined, it 
becomes very easy to compare them with the whole observed 
course of the comet, by a process exactly similar to that of 
art. 502, and thus at once to ascertain their correctness, and 
to put to the severest trial the truth of those general laws on 
which all such calculations are founded. 

(564.) For the most part, it is found that the motions of 
comets may be sufficiently well represented by parabolic 
orbits — that is to say, ellipses whose axes are of infinite 
length, or, at least, so very long that no appreciable error 
in the calculation of their motions, during all the time they 
continue visible, would be incurred by supposing them ac- 
tually infinite. The parabola is that conic section which 
is the limit between the ellipse on the one hand, which re- 
turns into itself, and the hyperbola on the other, which runs 
out to infinity. A comet, therefore, which should describe 
an elliptic path, however long its axis, must have visited the 
sun before, and must again return (unless disturbed) in some 
determinate period — but should its orbit be of the hyper- 
bolic character, when once it had passed its perihelion, it 
could never more return within the sphere of our observa- 
tion, but must run off to visit other systems, or be lost in 
the immensity of space. There is no instance of a comet 
whose orbit has been very carefully calculated by more than 
one computist being proved to have described a hyperbola, 



OUTLINES OF ASTRONOMY 475 

though several have been suspected of doing so. 6 Many 
have been well ascertained to move in ellipses. These lat- 
ter, in so far as their orbits can remain unaltered by the 
attractions of the planets, must be regarded as permanent 

members of our system. 

(665.) We mast now say a few words on the actual 
dimensions of comets. The calculation of the diameters 
of their heads, and the lengths and breadths of their tails, 
offers not the slightest difficulty when once the elements of 
their orbits are known, for by these we know their real dis- 
tances from the earth at any time, and the true direction of 
the tail, which we see only foreshortened. Now calcula- 
tions instituted on these principles lead to the surprising 
fact, that the comets are by far the most voluminous bodies 
in our system. The following are the dimensions of some 
of those which have been made the subjects of such inquiry. 

(566.) The tail of the great comet of 1680, immediately 
after its perihelion passage, was found by Newton to have 
been no less than 20000000 of leagues in length, and to have 
occupied only two days in its emission from the comet's 
body! a decisive proof this of its being darted forth by some 
active force, the origin of which, to judge from the direction 
of the tail, must be sought in the sun itself. Its greatest 
length amounted to 41000000 leagues, a length much exceed- 
ing the whole interval between the sun and earth. The tail 
of the comet of 1769 extended 16000000 leagues, and that of 
the great comet of 1811, 36000000. The portion of the head 
of this last, comprised within the transparent atmospheric 
envelope which separated it from the tail, was 180000 leagues 



6 For example that of 1723, calculated by Burckhardt; that of 1771, by 
both Burckhardt and Encke; and the second comet of 1818, by Rosenberg and 
Schwabe. 



476 OUTLINES OF ASTRONOMY 

in diameter. It is hardly conceivable, that matter once pro- 
jected to such enormous distances should ever be collected 
again by the feeble attraction of such a body as a comet — a con- 
sideration which accounts for the surmised progressive dimi- 
nution of the tails of such as have been frequently observed. 
(567.) The most remarkable of those comets which have 
been ascertained to move in elliptic orbits is that of Halley, 
so called from the celebrated Edmund Halley, who, on cal- 
culating its elements from its perihelion passage in 1682, 
when it appeared in great splendor, with a tail 30° in length, 
was led to conclude its identity with the great comets of 
1531 and 1607, whose elements he had also ascertained. The 
intervals of these successive apparitions being 75 and 76 
years, Halley was encouraged to predict its reappearance 
about the year 1759. So remarkable a prediction could not 
fail to attract the attention of all astronomers, and, as the 
time approached, it became extremely interesting to know 
whether the attractions of the larger planets might not mate- 
rially interfere with its orbitual motion. The computation 
of their influence from the Newtonian law of gravity, a most 
difficult and intricate piece of calculation, was undertaken 
and accomplished by Clairaut, who found that the action of 
Saturn would retard its return by 100 days, and that of 
Jupiter by no less than 518, making in all 618 days, by 
which the expected return would happen later than on the 
supposition of its retaining an unaltered period — and that, 
in short, the time of the expected perihelion passage would 
take place within a month, one way or other, of the middle 
of April, 1759.— It actually happened on the 12fch of March 
in that year. Its next return was calculated by several emi- 
nent geometers, 6 and fixed successively for the 4th, the 7th, 

6 Damoiseau, Pontecoulam, Rosenberger and Lelioiana. 



OUTLINES OF ASTRONOMY 477 

the 11th, and the 26th of November, 1835; the two latter 
determinations appearing entitled to the higher degree of 
contidence, owing partly to the more complete discussion 
bestowed on the observations of 1682 and 1759, and partly 
to the continually improving state of our knowledge of the 
methods of estimating the disturbing effect of the several 
planets. The last of these predictions, that of M. Lehmann, 
was published on the 25th of July. On the 5th of August 
the comet first became visible in the clear atmosphere of 
Eome as an exceedingly faint telescopic nebula, within a 
degree of its place as predicted by M. Eosenberger for that 
day. On or about the 20th of August it became generally 
visible, and, pursuing very nearly its calculated path among 
the stars, passed its perihelion on the 16th of November; 
after which, its course carrying it south, it ceased to be visi- 
ble in Europe, though it continued to be conspicuously so 
in the southern hemisphere throughout February, March, 
and April, 1836, disappearing finally on the 5th of May. 

(568.) Although the appearance of this celebrated comet 
at its last apparition was not such as might be reasonably 
considered likely to excite lively sensations of terror, even 
in superstitious ages, yet, having been an object of the most 
diligent attention in all parts of the world to astronomers, 
furnished with telescopes very far surpassing in power those 
which had been applied to it at its former appearance in 
1759, and indeed to any of the greater comets on record, the 
opportunity thus afforded of studying its physical structure, 
and the extraordinary phenomena which it presented when 
so examined have rendered this a memorable epoch in com- 
etic history. Its first appearance, while yet very remote 
from the sun, was that of a small round or somewhat oval 
nebula, quite destitute of tail, and having a minute point of 



478 OUTLINES OF ASTRONOMY 

more concentrated light excentrically situated within it. It 
was not before the 2d of October that the tail began to 
be developed, and thenceforward increased pretty rapidly, 
being already 4° or 5° long on the 5th. It attained its 
greatest apparent length (about 20°) on the 15th of October. 
From that time, though not yet arrived at its perihelion, it 
decreased with such rapidity, that already on the 29th it was 
only 3°, and on November the 5th 2i° in length. There is 
every reason to believe that before the perihelion, the tail had 
altogether disappeared, as, though it continued to be observed 
at Pulkowa up to the very day of its perihelion passage, no 
mention whatever is made of any tail being then seen. 

(569.) By far the most striking phenomena, however, 
observed in this part of its career, were those which, com- 
mencing simultaneously with the growth of the tail, con- 
nected themselves evidently with the production of that 
appendage and its projection from the head. On the 2d of 
October (the very day of the first observed commencement 
of the tail) the nucleus, which had been faint and small, 
was observed suddenly to have become much brighter, and 
to be in the act of throwing out a jet or stream of light from 
its anterior part, or that turned toward the sun. This ejec- 
tion after ceasing a while was resumed, and with much 
greater apparent violence, on the 8th, and continued, with 
occasional intermittences, so long as the tail itself continued 
visible. Both the form of this himinous ejection, and the 
direction in which it issued from the nucleus, meanwhile 
underwent singular and capricious alterations, the different 
phases succeeding each other with such rapidity that on no 
two successive nights were the appearances alike. At one 
time the emitted jet was single, and confined within narrow 
limits of divergence from the nucleus. At others it pre- 



OUTLINES OF ASTRONOMY 479 

sented a fan-shaped or swallow-tailed form, analogous to 
that of a gas-flame issuing from a flattened orifice: while 
at others again two, three or even more jets were darted 
forth in different directions. 7 (See figures a, 6, c, d, Plate 
I. fig. 4, which represent, highly magnified, the appearances 
of the nucleus with its jets of light, on the 8th, 9th, 10th, 
and 12th of October, and in which the direction of the 
anterior portion of the head, or that fronting the sun, is 
supposed alike in all, viz. toward the upper part of the 
engraving. In these representations the head itself is 
omitted, the scale of the figures not permitting its intro- 
duction: e represents the nucleus and head as seen October 
9th on a less scale.) The direction of the principal jet was 
observed meanwhile to oscillate to and fro on either side 
of a line directed to the sun in the manner of a compass- 
needle when thrown into vibration and oscillating about a 
mean position, the change of direction being conspicuous 
even from hour to hour. These jets, though very bright at 
their point of emanation from the nucleus, faded rapidly 
away, and became diffused as they expanded into the coma, 
at the same time curving backward as streams of steam or 
smoke would do, if thrown out from narrow orifices, more 
or less obliquely in opposition to a powerful wind, against 
which they were unable to make way, and, ultimately yield- 
ing to its force, so as to be drifted back and confounded in a 
vaporous train, following the general direction of the current. 8 
(570.) Eeflecting on these phenomena, and carefully con- 

1 See the exquisite lithographic representations of these phenomena by Bes- 
sel, Astron. Nachr. No. 302, and the fine series by Schwabe in No. 297 of 
that collection, as also the magnificent drawings of Struve, from which our 
figures a, b, c, d, are copied. 

8 On this point Schwabe's and Bessel's drawings are very express and 
unequivocal. Struve's attention seems to have been more especially directed 
to the scrutiny of the nucleus. 



480 OUTLINES OF ASTRONOMY 

sidering the evidence afforded by the numerous and elabo- 
rately executed drawings which have been placed on record 
by observers, it seems impossible to avoid the following 
conclusions. 1st. That the matter of the nucleus of a comet 
is powerfully excited and dilated into a vaporous state by 
the action of the sun's rays, escaping in streams and jets 
at those points of its surface which oppose the least resist- 
ance, and in all probability throwing that surface or the 
nucleus itself into irregular motions by its reaction in the 
act of so escaping, and thus altering its direction. 

2dly. That this process chiefly takes place in that por- 
tion of the nucleus which is turned toward the sun; the 
vapor escaping chiefly in that direction. 

3dly. That when so emitted, it is prevented from pro- 
ceeding in the direction originally impressed upon it, by 
some force directed from the sun, drifting it back and carry- 
ing it out to vast distances behind the nucleus, forming the 
tail or so much of the tail as can be considered as consisting 
of material substance. 

4thly. That this force, whatever its nature, acts un- 
equally on the materials of the comet, the greater portion 
remaining unvaporized, and a considerable part of the 
vapor actually produced, remaining in its neighborhood, 
forming the head and coma. 

5thly. That the force thus acting on the materials of the 
tail cannot possibly be identical with the ordinary gravita- 
tion of matter, being centrifugal or repulsive, as respects 
the sun, and of an energy very far exceeding the gravitating 
force toward that luminary. This will be evident if we con- 
sider the enormous velocity with which the matter of the 
tail is carried backward, in opposition both to the motion 
which it had as part of the nucleus, and to that which it 



OUTLINES OF ASTRONOMY 481 

acquired in the act of its emission, both which motions 
have to be destroyed in the first instance, before any move- 
ment in the contrary direction can be impressed. 

6thly. That unless the matter of the tail thus repelled 
from the sun be retained by a peculiar and highly energetic 
attraction to the nucleus, differing from and exceptional 
to the ordinary power of gravitation, it must leave the 
nucleus altogether; being in effect carried far beyond the 
coercive power of so feeble a gravitating force as would 
correspond to the minute mass of the nucleus; and it is 
therefore very conceivable that a comet may lose, at every 
approach to the sun, a portion of that peculiar matter, 
whatever it be, on which the production of its tail depends, 
the remainder being of course less excitable by the solar 
action, and more impassive to his rays, and therefore, pro 
tanto, more nearly approximating to the nature of the 
planetary bodies. 

7thly. That considering the immense distances to which 
at least some portion of the matter of the tail is carried from 
the comet, and the way in which it is dispersed through the 
system, it is quite inconceivable that the whole of that 
matter should be reabsorbed — that therefore it must lose 
during its perihelion passage some portion of its matter, 
and if, as would seem far from improbable, that matter 
should be. of a nature to be repelled from, not attracted by, 
the sun, the remainder will, by consequence, be, pro quan- 
titate inertice, more energetically attracted to the sun than the 
mean of both. If then the orbit be elliptic, it will perform 
each successive revolution in a shorter time than the pre- 
ceding, until, at length, the whole of the repulsive matter 
is got rid of. — But to return to the comet of Halley. 

(571.) After the perihelion passage, the comet was lost 



482 OUTLINES OF ASTRONOMY 

sight of for upward of two months, and at its reappearance 
(on the 24th of January, 1836) presented itself under quite 
a different aspect, having in the interval evidently under- 
gone some great physical change which had operated an 
entire transformation in its appearance. It no longer pre- 
sented any vestige of tail, but appeared to the naked eye 
as a hazy star of about the fourth or fifth magnitude, and 
in powerful telescopes as a small, round, well defined disk, 
rather more than 2' in diameter, surrounded with a nebulous 
chevelure or coma of much greater extent. Within the disk, 
and somewhat excentrically situated, a minute but bright 
nucleus appeared, from which extended toward the pos- 
terior edge of the disk (or that remote from the sun) a short 
vivid luminous ray. (See Jig. 1, Plate VI.) As the comet 
receded from the sun, the coma speedily disappeared, as if 
absorbed into the disk, which, on the other hand, increased 
continually in dimensions, and that with such rapidity, that 
in the week elapsed from January 25th to February 1st 
(calculating from micrometrical measures, and from tks 
known distance of the comet from the earth on these 
days), the actual volume or real solid content of the illumi- 
nated space had dilated in the ratio of upward of 40 to I. 9 
And so it continued to swell out with undiminished rapid- 
ity, until, from this cause alone, it ceased to be visible, the 
illumination becoming fainter as the magnitude increased; 
till at length the outline became indistinguishable from 

9 On the night of the 22d of January the comet was observed by M. Bogus- 
law ski of Breslau, as a star of the sixth magnitude, a bright concentrated point; 
which showed no disk with a magnifying power of 140, and that it actually was 
the comet he assured himself by turning his telescope the next night on the 
place where he saw it (which he had carefully noted and registered). It was 
gone. From his observation it appears then that at I7h 50m M. T. at, Breslau, 
Jan. 22, the diameter of the nucleus with its envelope was rigorously nil, at 
which moment, within an hour one way or the other, the process of formation 
of the envelope must have commenced. 



OUTLINES OF ASTRONOMY 483 

simple want of light to trace it. While this increase of 
dimension proceeded, the form of the disk passed, by- 
gradual and successive additions to its length in the direc- 
tion opposite to the sun, to that of a paraboloid, as repre- 
sented in #, Jig. 1, Plate VI., the anterior curved portion 
preserving its planetary sharpness, but the base being faint 
and ill-defined. It is evident that had this process con- 
tinued with sufficient light to render the result visible, a 
tail would have been ultimately reproduced; but the in- 
crease of dimension being accompanied with diminution of 
brightness, a short, imperfect, and as it were rudimentary 
tail only was formed, visible as such for a few nights to 
the naked eye, or in a low magnifying telescope, and that 
only when the comet itself had begun to fade away by 
reason of its increasing distance. 

(572.) While the parabolic envelope was thus contin- 
ually dilating and growing fainter, the nucleus underwent 
little change, but the ray proceeding from it increased in 
length and comparative brightness, preserving all the time 
its direction along the axis of the paraboloid, and offering 
none of those irregular and capricious phenomena which 
characterized the jets of light emitted anteriorly, previous 
to the perihelion. If the office of those jets was to feed the 
tail, the converse office of conducting back its successively 
condensing matter to the nucleus would seem to be that of 
the ray now in question. By degrees this also faded, and 
the last appearance presented by the comet was that which 
it offered at its first appearance in August, viz. that of a 
small round nebula with a bright point in or near the centre. 

(573.) Besides the comet of H alley, several other of the 

great comets recorded in history have been surmised with 

more or less probability to return periodically, and therefore 

Astronomy— Vol. XX— 2 



484 OUTLINES OF ASTRONOMY 

to move in elongated ellipses around the sun. Such is the 
great comet of 1680, whose period is estimated at 575 years, 
and which has been considered, with at least a high prima 
facie probability, to be identical with a magnificent comet 
observed at Constantinople and in Palestine, and referred 
by contemporary historians, both European and Chinese, to 
the year A.D. 1106; with that of A.D. 531, which was seen 
at noonday close to the sun; with the comet of 43 B.C., 
already spoken of as having appeared after the death of 
Caesar, and which was also observed in the daytime; and 
finally with two other comets, mention of which occurs in 
the Sibylline Oracles, and in a passage of Homer, and 
which are referred, as well as the obscurity of chronology 
and the indications themselves will allow, to the years 618 
and 1194 B.C. It is to the assumed near approach of this 
comet to the earth about the time of the Deluge, that Whis- 
ton ascribed that overwhelming tide-wave to whose agency 
his wild fancy ascribed that great catastrophe — a specula- 
tion, it is needless to remark, purely visionary. These co- 
incidences of time are certainly remarkable, especially when 
it is considered how very rare are the appearances of comets 
of this class. Professor Encke, however, has discussed, 
with all possible care, the observations recorded of the 
comet of 1680, taking into consideration the perturbations 
of the planets (which are of trifling importance, by reason 
of the great inclination of its orbit to the ecliptic), and his 
calculations show that no elliptic orbit, with such a period 
as 575 years, is competent to represent them within any 
probable or even possible limits of error, the most probable 
period assigned by them being 8814 Julian years. Inde- 
pendent of this consideration, there are circumstances re- 
corded of the comet of A.D. 1106 incompatible with its 



OUTLINES OF ASTRONOMY 485 

motion in any orbit identical with that of the comet of 1680, 
so that the idea of referring all these phenomena to one and 
the same comet, however seducing, must be relinquished. 

(574.) Another great comet, whose return about the year 
1848 had been considered by more than one eminent author- 
ity in this department of astronomy 10 highly probable, is 
that of 1556, to the terror of whose aspect some historians 
have attributed the abdication of the Emperor Charles V. 
This comet is supposed to be identical with that of 1264, 
mentioned by many historians as a great comet, and ob- 
served also in China — the conclusion in this case resting 
upon' the coincidence of elements calculated on the obser- 
vations, such as they are, which have been recorded. On 
the subject of this coincidence Mr. Hind has entered into 
many elaborate calculations, the result of which is strongly 
in favor of the supposed identity. This probability is fur- 
ther increased by the fact of a comet, with a tail of 40° and 
a head bright enough to be visible after sunrise, having 
appeared in A.D. 975: and of two others having been re- 
corded by the Chinese annalists in A.D. 395 and 104. It 
is true that if these be the same, the mean period would 
be somewhat short of 292 years. But the effect of planet- 
ary perturbation might reconcile even greater differences^ 
and though up to the time of our writing (1858) no such 
comet has yet been observed, two or three years must yet 
elapse, in the opinion of those best competent to judge, 
before its return must be considered hopeless. 

(575.) In 1661, 1532, 1402, 1145, 891 and 243 great com- 
ets appeared — that of 1402 being bright enough to be seen 
at noonday. A period of 129 years would conciliate all 
these appearances, and should have brought back the comet 



10 Pingre, Cometographie, i. 411; Lalande, Astr. 3185. 



486 OUTLINES OF ASTRONOMY 

in 1789 or 1790 (other circumstances agreeing). That no 
such comet was observed about that time is no proof that 
it did not return, since, owing to the situation of its orbit, 
had the perihelion passage taken place in July it might 
have escaped observation. Mechain, indeed, from an elab- 
orate discussion of the observations of 1532 and 1661, came 
to the conclusion that these comets were .not the same; but 
the elements assigned by Olbers to the earlier of them differ 
so widely from those of Mechain for the same comet on the 
one hand, and agree so well with those of the last named 
astronomer for the other, 11 that we are perhaps justified in 
regarding the question as not yet set at rest. 

(576.) We come now, however, to a class of comets of 
short period, respecting whose return there is no doubt, 
inasmuch as two at least of them have been identified as 
having performed successive revolutions round the sun; 
have had their return predicted already several times; and 
have on each occasion scrupulously kept to their appoint- 
ments. The first of these is the comet of Encke, so called 
from Professor Encke of Berlin, who first ascertained its 
periodical return. It revolves in an ellipse of great excen- 
tricity (though not comparable to that of Halley's), the 
plane of which is inclined at an angle of about 13° 22' to 
the plane of the ecliptic, and in the short period of 1211 
days, or about 3 J years. This remarkable discovery was 
made on the occasion of its fourth recorded appearance, in 
1819. From the ellipse then calculated by Encke, its re- 
turn in 1822 was predicted by him, and observed at Para- 
matta, in New South Wales, by M. Eiimker, being invisi- 
ble in Europe: since which it has been repredicted and 
reobserved in all the principal observatories, both in the 

11 See Schumacher's Catal. Astron. Abhandl. i. 



OUTLINES OF ASTRONOMY 487 

northern and southern hemispheres, as a phenomenon of 
regular occurrence. 

(577.) On comparing the intervals between the successive 
perihelion passages of this comet, after allowing in the most 
careful and exact manner for all the disturbances due to the 
actions of the planets, a very singular fact has come to 
light, viz. that the periods are continually diminishing, or, 
in other words, the mean distance from the sun, or the 
major axis of the ellipse, dwindling by slow and regular 
degrees at the rate of about d -11176 per revolution. This 
is evidently the effect which would be produced by a re- 
sistance experienced by the comet from a very rare ethereal 
medium pervading the regions in which it moves; for such 
resistance, by diminishing its actual velocity, would dimin- 
ish also its centrifugal force, and thus give the sun more 
power over it to draw it nearer. Accordingly this is the 
solution proposed by Encke, and at present generally re- 
ceived. Should this be really the case, it will, therefore, 
probably fall ultimately into the sun, should it not first be 
dissipated altogether — a thing no way improbable, when the 
lightness of its materials is considered. The considerations 
adduced at the end of art. 570 would seem, however, to 
open out another possible explanation of the phenomenon 
in question, not necessarily leading to such a catastrophe. 

(578.) By measuring the apparent magnitude of this 
comet at different distances from the sun, and thence, 
from a knowledge of its actual distance from the earth 
at the time, concluding its real volume, it has been ascer- 
tained to contract in bulk as it approaches to, and to ex- 
pand as it recedes from, that luminary. M. "Valz, who was 
the first to notice this fact, accounts for it by supposing it 
to undergo a real compression or condensation of volume 



488 OUTLINES OF ASTRONOMY 

arising from the pressure of an ethereal medium which he 
conceives to grow more dense in the sun's neighborhood. 
But such a hypothesis is evidently inadmissible, since it 
would require us to assume the exterior of the comet to 
be iu the nature of a skin or bag impervious to the com- 
pressing medium. The phenomenon is analogous to the 
increase of dimension above described, as observed in the 
comet of Halley, when in the act of receding from the sun, 
and is doubtless referable to a similar cause, viz. the alter- 
nate conversion of evaporable matter into the states of visi- 
ble cloud and invisible gas, by the alternating action of 
cold and heat. This cornet has no tail, but offers to the 
view only a small ill-denned nucleus, excentrically situated 
within a more or less elongated oval mass of vapors, being 
nearest to that vertex which is toward the sun. 

(579.) Another comet of short period is that of Biela, 
so called from M. Biela, of Josephstadt, who first arrived 
at this interesting conclusion on the occasion of its appear- 
ance in 1826. It is considered to be identical with comets 
which appeared in 1772, 1805, etc., and describes its very 
excentric ellipse about the sun in 2410 days, or about 
6| years; and in a plane inclined 12° 34' to the ecliptic. 
It appeared again, according to the prediction, in 1832 
and in 1846. Its orbit, by a remarkable coincidence, very 
nearly intersects that of the earth; and had the latter, at 
the time of its passage in 1832, been a month in advance 
of its actual place, it would have passed through the 
comet — a singular rencontre, perhaps not unattended with 
danger. 12 

12 Should calculation establish the fact of a resistance experienced also by 
this comet, the subject of periodical comets will assume an extraordinary degree 
of interest. It cannot be doubted that many more will be discovered, and by 
their resistance questions will come to be decided, such as the following: — What 



OUTLINES OF ASTRONOMY 489 

(580.) This comet is small and hardly visible to the 
naked eye, even when brightest. Nevertheless, as if to 
make up for its seeming insignificance by the interest 
attaching to it in a physical point of view, it exhibited, 
at its appearance in 1846, a phenomenon which struck 
every astronomer with amazement, as a thing without 
previous example in the history of our system. 13 It was 
actually seen to separate itself into two distinct comets, 
which, after thus parting company, continued to journey 
along amicably through an arc of upward of 70° of their 
apparent orbit, keeping all the while within the same field 
of view of the telescope pointed toward them. The first 
indication of something unusual being about to take place 
might be, perhaps, referred to the 19th of December, 1845, 
when the comet appeared to Mr. Hind pear-shaped, the 
nebulosity being unduly elongated in a direction inclining 
northward. But on the 13th of January, at Washington in 
America, and on the 15th. and subsequently in every part 
of Europe, it was distinctly seen to have become double; a 
very small and faint cometic body, having a nucleus of its 
own, being observed appended to it, at a distance of about 



is the law of density of the resisting medium which surrounds the sun? Is it 
at re3t or in motion? If the latter, in what direction does it move? Circularly 
round the sun, or traversing space? If circularly, in what plane? It is obvious 
that a circular or vorticose motion of the ether would accelerate some comets and 
retard others, according as their revolution was, relative to such motion, direct 
or retrograde. Supposing the neighborhood of the sun to be rilled with a ma- 
terial fluid, it is not conceivable that the circulation of the planets in it for ages 
should not have impressed upon it some degree of rotation in their own direc- 
tion. And this may preserve them from the extreme effects of accumulated 
resistance. — Note to edition o/1833. 

13 Perhaps not quite so. To say nothing of a singular surmise of Kepler, 
that two great comets seen at once in 1618, might be a single comet separated 
into two, the following passage of Hevelius cited by M. Littrow (Nachr. 564) 
does really seem to refer to some phenomenon bearing at least a certain analogy 
to it. "In ipso disco," he says (Cometographia, p. 326), "quatuor vel quinque 
corpus cula qusedam sive nucleos reliquo corpore aliquanto densiores osten- 
debat." 



490 OUTLINES OF ASTRONOMY 

2 ; (in arc) from its centre, and in a direction forming an 
angle of about 328° with the meridian, running northward 
from the principal or original comet (see art 204). From 
this time the separation of the two comets west on pro- 
gressively, though slowly. On the 30th of January, the 
apparent distance of the nucleus had increased to 3', on 
the 7th of February to 4', and on the 18th to 5', and so 
on, until on the 5th of March the two comets were sepa- 
rated by an interval of 9' 19", the apparent direction of 
the line of junction all the while varying but little with 
respect to the parallel. 14 

(581.) During this separation, very remarkable changes 
were observed to be going on, both in the original comet 
and its companion. Both had nuclei, both had snort tails, 
parallel in direction, and nearly perpendicular to the line 
of junction, but whereas at its first observation on January 
13th, the new comet was extremely small and faint in com- 
parison with the old, the difference both in point of light 
and apparent magnitude diminished. On the 10th of Feb- 
ruary, they were nearly equal, although the day before the 
moonlight had effaced the new one, leaving the other bright 
enough to be well observed. On the 14th and 16th, how- 
ever, the new comet had gained a decided superiority of 
light over the old, presenting at the same time a sharp and 
starlike nucleus, compared by Lieut. Maury to a diamond 
spark. But this state of things was not to continue. Al- 
ready, on the 18th, the old comet had regained its superi- 
ority, being nearly twice as bright as its companion, and 
offering an unusually bright and starlike nucleus. From 



i4 By far the greater portion of this increase of apparent distance was dua 
to the comet's increased proximity to the earth. The real iucrease reduced to 
a distance = 1 of the comet was at the rate of about 3" per diem. 



OUTLINES OF ASTRONOMY 491 

this period the new companion began to fade away, but 
continued visible up to the 15th of March. On the 24th 
the comet appeared again single, and on the 22d of April 
both had disappeared. 

(582.) While this singular interchange of light was going 
forward, indications of some sort of communication between 
the comets were exhibited. The new or companion comet, 
besides its tail, extending in a direction parallel to that of 
the other, threw out a faint arc of light which extended as 
a kind of bridge from the one to the other; and after the 
restoration of the original comet to its former pre-eminence, 
it, on its part, threw forth additional rajs, so as to present 
(on the 22d and 23d February) the appearance of a comet 
with three faint tails forming angles of about 120° with each 
other, one of which extended toward its companion. 

(583.) Professor Plantamour, director of the observatory 
of Geneva, having investigated the orbits of both these 
comets as separate and independent bodies, from the exten- 
sive and careful series of observations made upon them, ar- 
rived at the conclusion that the increase of distance between 
the two nuclei, at least during the interval from February 10th 
to March 22d, was simply apparent, being due to the varia- 
tion of distance from the earth, and to the angle under which 
their line of junction presented itself to the visual ray; the 
real distance during all that interval (neglecting small frac- 
tions) having been on an average about thirty-nine times the 
semidiameter of the earth, or less than two-thirds the dis- 
tance of the moon from its centre. From this it would ap- 
pear that already, at this distance, the two bodies had ceased 
to exercise any perceptible amount of perturbative gravita- 
tion on each other; as, indeed, from the probable minute- 
ness of cometary masses we might reasonably expect. It 



492 OUTLINES OF ASTRONOMY 

may well be supposed that astronomers would not allow so 
remarkable a duplication to pass unwatched at the next re- 
turn of the comet in 1852. In August and September of 
that year both nuclei were observed by Professor Challis at 
Cambridge, Secchi at Eome, and M. Struve, presenting, as 
regards direction, the same relative situation with regard to 
each other, so that we have here the historical proof of a 
permanent addition to the members of our system taking 
place at a definite instant under our very eye. 16 (Plate VI. 
fig. 2.) 

(584.) A third comet, of short period, has been added to 
our list by M. Faye, of the observatory of Paris, who de- 
tected it on the 22d of November, 1843. A very few obser- 
vations sufficed to show that no parabola would satisfy tho 
conditions of its motion, and that to represent them com- 
pletely, it was necessary to assign to it an elliptic orbit of 
very moderate excentricity. The calculations of M. Nicolai, 
subsequently revised and slightly corrected by M. Lever- 
rier, have shown that an almost perfect representation of its 
motions during the whole period of its visibility would be 
afforded by assuming it to revolve in a period of 2717 d *68 (or 
somewhat less than 7£ years) in an ellipse whose excentricity 
is 0-55596, and inclination to the ecliptic 11° 22' 31"; and 
taking this for a basis of further calculation, and by means 
of these data and the other elements of the orbit estimating 
the effect of planetary perturbation during the revolution 
now in progress, he fixed its next return to the perihelion 
for the 3d of April, 1851, with a probable error one way or 

lj According to the elements of this comet deduced by M. Santini, taking 
into account all planetary perturbation, its two heads ought to have passed their 
perihelion on January 27 and January 29, respectively, 1866. Its appearance 
was anxiously and persevering! y looked for, but in vain; nor has ony probable 
cause been assigned for its disappearance] 



OUTLINES OF ASTRONOMY 493 

other not exceeding one or two days. This prediction has 
been strikingly verified. It actually passed its perihelion 
on the 1st of April, 1851, having been rediscovered by Pro- 
fessor Challis at Cambridge in November, 1850, and fol- 
lowed beyond the perihelion by M. Otto Struve up to March 
4, 1851. 

(585.) The effect of planetary perturbation on the motion 
of comets has been more than once alluded to in what has 
been above said. Without going minutely into this part of 
the subject, which will be better understood after the perusal 
of a subsequent chapter, it must be obvious, that as the 
orbits of comets are very excentric, and inclined in all sorts 
of angles to the ecliptic, they must, in many instances, if 
not actually intersect, at least pass very near to the orbits of 
some of the planets. We have already seen, for instance, 
that the orbit of Biela's comet so nearly intersects that of 
the earth, that an actual collision is not impossible, and in- 
deed (supposing neither orbit variable) must in all likelihood 
happen in the lapse of some millions of years. Neither are 
instances wanting of comets having actually approached the 
earth within comparatively short distances, as that of 1770, 
which on the 1st of July of that year was within little more 
than seven times the moon's distance. The same comet, in 
1767, passed Jupiter at a distance only one 58th of the radius 
of that planet's orbit, and it has been rendered extremely 
probable that it is to the disturbance its former orbit under- 
went during that appulse that we owe its appearance within 
our own range of vision. This exceedingly remarkable 
comet was found by Lexell to describe an elliptic orbit with 
an excentricity of 0*7858, with a periodic time of about five 
years and a half, and in a plane only 1° 34' inclined to the 
ecliptic, having passed its perihelion on the 13th of August, 



494 OUTLINES OF ASTRONOMY 

1770. Its return of coarse was eagerly expected, but in 
vain, for the comet has never been certainly identified with 
any comet since seen. Its observation on its first return in 
1776 was rendered impossible by the relative situations of 
the perihelion and of the earth at the time, and before an- 
other revolution could be accomplished (as has since been 
ascertained), viz. about the 23d of August, 1779, by a sin- 
gular coincidence it again approached Jupiter within one 
491st part of its distance from the sun, being nearer to that 
planet by one-fifth than its tourth satellite. No wonder, 
therefore, that the planet's attraction (which at that distance 
would exceed that of the sun in the proportion of at least 
200 to 1) should completely alter the orbit and deflect it into 
a curve, not one of whose elements would have the least 
resemblance to those of the ellipse of Lexell. It is worthy of 
notice that by this rencontre with the system of Jupiter's 
satellites, none of their motions suffered any perceptible de- 
rangement — a sufficient proof of the smallness of its mass. 
Jupiter, indeed, seems, by some strange fatality, to be con- 
stantly in the way of comets, and to serve as a perpetual 
stumbling-block to them. 

(586.) On the 22d of August, 1844, Signor de Vico, 
director of the observatory of the Oollegio Romano, discov- 
ered a comet, the motions of which, a very few observations 
sufficed to show, deviated remarkably from a parabolic orbit. 
It passed its perihelion on the 2d of September, and contin- 
ued to be observed until the 7th of December. Elliptic ele- 
ments of this comet, agreeing remarkably well with each 
other, were accordingly calculated by several astronomers, 
from which it appears that the period of revolution is about 
1990 days, or 5£ (5*4357) years, which (supposing its orbit 
undisturbed in the interim) would bring it back to the peri- 



OUTLINES OF ASTRONOMY 495 

lielion 011 or about the 13th of January, 1850, on which occa- 
sion, however, by reason of its unfavorable situation with 
respect to the sun and earth, it could not be observed. As 
the assemblage and comparison of these elements, thus com- 
puted independently, will serve better, perhaps, than any 
other example, to afford the student an idea of the degree of 
arithmetical certainty capable of being attained in this 
branch of astronomy, difficult and complex as the calcula- 
tions themselves are, and liable to error as individual obser- 
vations of a body so ill defined as the smaller comets are for 
the most part, we shall present them in a tabular form, as 
on the next page: the elements being as usual; the time of 
perihelion passage, longitude of the perihelion, that of the 
ascending node, the inclination to the ecliptic, semiaxis and 
excentricity of the orbit, and the periodic time, 

This comet, when brightest, was visible to the naked 
eye, and had a small tail. It is especially interesting to 
astronomers from the circumstance of its having been ren- 
dered exceedingly probable by the researches of M. Lever- 
rier, that it is identical with one which appeared in 1678 
with some of its elements considerably changed by pertur- 
bation. This comet is further remarkable, from having 
been concluded by Messrs. Laugier and Mauvais, to be iden- 
tical with the comet of 1585 observed by Tycho Brahe, and 
possibly also with those of 1743, 1766 and 1819. 

(587.) Elliptic elements have in like manner been as- 
signed to the comet discovered by M. Brorsen, on the 26th 
of February, 1846, which, like that last mentioned, speedily 
after its discovery began to show evident symptoms of de- 
viation from a parabola. These elements, with the names 
of their respective calculators, are as follows. The dates are 
for February, 1846, Greenwich time. 



496 



OUTLINES OF ASTRONOMY 



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OUTLINES OF ASTRONOMY 



497 



e& by 



Perihe: 

Long, of Perihelion 

Loup, of £, 

Inclination 

Semi axis . 
Eccentricity 
Period (days) 



1 

Bru: 


25d -37794 

116° 28' 34" 

102 39 36- 

30 55 6- 

3-15021 

0-79363 

2042 


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25 d -33109 

116°28'17"S 

102 45 20- 9 

30 49 3- 6 

312292 

0-79771 

2016 



Van Willingen 


and De Haan. 


23' 1 -02227 


116°23'52"-9 


103 31 25- 7 


30 30 30- 2 


2-87052 


0-77313 


1T76 



This comet is faint, and presents nothing remarkable in 
its appearance. Its chief interest arises from the great simi- 
larity of its parabolic elements to those of the comet of 1532, 
the place of the perihelion and node, and the inclination of 
the orbit, being almost identical. 

(588.) Elliptic elements have also been calculated by M. 
D' Arrest, for a comet discovered by M. Peters, on the 26th 
of June, 1846, which go to assign it a place among the com- 
ets of short period, viz. 5804 d *3, or very nearly 16 years. 
The excentricity of the orbit is 0*75672, its semiaxis 6*32066, 
and the inclination of its plane to that of the ecliptic 31° 
2' 14". This comet passed its perihelion on the 1st of June, 
1846. 

(589.) By far trie most remarkable comet, however, which 
has been seen during the present century, is that which ap- 
peared in the spring of 1843, and whose tail became visible 
in the twilight of the 17th of March in England as a great 
beam of nebulous light, extending from a point above the 
western horizon, through the stars of Eridanus and Lepus, 
under the belt of Orion. This situation was low and un- 
favorable; and it was not till the 19th that the head was 
seen, and then only as a faint and ill-defined nebula, very 
rapidly fading on subsequent nights. In more southern 
latitudes, however, not only the tail was seen, as a magnifi- 



498 OUTLINES OF ASTRONOMY 

cent train of light extending 50° or 60° in length; but the 
head and nucleus appeared with extraordinary splendor, 
exciting in every country where it was seen the greatest 
astonishment and admiration. Indeed, all descriptions agree 
in representing it as a stupendous spectacle, such as in 
superstitious ages would not fail to have carried terror 
into every bosom. In tropical latitudes in the northern 
hemisphere, the tail appeared on the 3d of March, and in 
Yan Diemen's Land, so early as the 1st, the comet having 
passed its perihelion on the 27th of February. Already on 
the 3d the head was so far disengaged from the immediate 
vicinity of the sun, as to appear for a short time above the 
horizon after sunset. On this day when viewed through a 
46-inch achromatic telescope it presented a planetary disk, 
from which rays emerged in the direction of the tail. The 
tail was double, consisting of two principal lateral streamers, 
making a very small angle with each other, and divided by 
a comparatively dark line, of the estimated length of 25°, 
prolonged however on the north side by a divergent 
streamer, making an angle of 5° or 6° with the general 
direction of the axis, and traceable as far as 65° from the 
head. A similar though fainter lateral prolongation ap- 
peared on the south side. A fine drawing of it of this 
date by C. P. Smyth, Esq., of the Eoyal Observatory, 
C.G.H., represents it as highly symmetrical, and gives the 
idea of a vivid cone of light, with a dark axis, and nearly 
rectilinear sides, inclosed in a fainter cone, the sides of 
which curve slightly outward. The light of the nucleus at 
this period is compared to that of a star of the first or second 
magnitude; and on the 11th, of the third; from which time 
it degraded in light so rapidly, that on the 19th it was in- 
visible to the naked eye, the tail all the while continuing 



OUTLINES OF ASTRONOMY 499 

brilliantly visible, though much more so at a distance from 
the nucleus, with which, indeed, its connection was not 
then obvious to the unassisted sight — a singular feature 
in the history of this body. The tail, subsequent to the 
3 d, was generally speaking a single straight or slightly 
curved broad band of light, but on the 11th it is recorded 
by Mr. Clerihew, who observed it at Calcutta, to have shot 
forth a lateral tail nearly twice as long as the regular one 
but fainter, and making an angle of about 18° with its direc- 
tion on the southern side. The projection of this ray (which 
was not seen either before or after the day in question) to so 
enormous a length (nearly 100°) in a single day conveys an 
impression of the intensity of the forces acting to produce 
such a velocity of material transfer through space, such as 
no other natural phenomenon is capable of exciting. It is 
clear that if we have to deal here with matter, such as we con- 
ceive it, viz. possessing inertia — at all, it must be under the 
dominion of forces incomparably more energetic than gravi- 
tation, and of quite a different nature. 

(590.) There is abundant evidence of the comet in ques- 
tion having been seen in full daylight, and in the sun's 
immediate vicinity. It was so seen on the 28th of February, 
the day after its perihelion passage, by every person on 
board the H.E.I.C.S. "Owen Glendower," then off the 
Cape, as a short dagger-like object close to the sun a little 
before sunset. On the same day at 3 h 6 m P.M., and conse- 
quently in full sunshine, the distance of the nucleus from 
the sun was actually measured with a sextant by Mr. Clarke 
of Portland, United States, the distance centre from centre 
being then only 3° 50' 43". He describes it in the following 
terms: "The nucleus and also every part of the tail were 
as well defined as the moon on a clear day. The nucleus 



500 



OUTLINES OF ASTRONOMY 



and tail bore the same appearance, and resembled a perfectly 
pure white cloud without any variation, except a slight 
change near the head, just sufficient to distinguish the 
nucleus from the tail at that point." The denseness of the 
nucleus was so considerable, that Mr. Clarke had no doubt 
it might have been visible upon the sun's disk, had it passed 
between that and the observer. The length of the visible 
tail resulting from these measures was 59' or not far from 
double the apparent diameter of the sun; and as we shall 
presently see that on the day in question the distance from 
the earth of the sun and comet must have been very nearly 
equal, this gives us about 1700000 miles for the linear 
dimensions of this the densest portion of that appendage, 
making no allowance for the foreshortening, which at that 
time was very considerable. 

(591.) The elements of this comet are among the most 
remarkable of any recorded. They have been calculated by 
several eminent astronomers, among whose results we shall 
specify only those which agree best; the earlier attempts to 
compute its path having been rendered uncertain by the 
difficulty attending exact observations of it in the first part 
of its visible career. The following are those which seem 
entitled to most confidence: 




(592.) What renders these elements so remarkable is the 
smallncss of the perihelion distance. Of all comets which 



OUTLINES OF ASTRONOMY 501 

have been recorded this has made the nearest approach to 
the sun. The sun's radius being the sine of his apparent 
semidiameter (16' 1") to a radius equal to the earth's mean 
distance = 1, is represented on that scale by 0-00466, which 
falls short of 0-00534, the perihelion distance found by 
taking a mean of all the foregoing results, by only 0*00068, 
or about one-seventh of its whole magnitude. The comet, 
therefore, approached the luminous surface of the sun within 
about a seventh part of the sun's radius! It is worth while 
to consider what is implied in such a fact. In the first 
place, the intensity both of the light and radiant heat of 
the sun at different distances from that luminary increase 
proportionally to the spherical area of the portion of the 
visible hemisphere covered by the sun's disk. This disk, 
in the case of the earth, at its mean distance has an angular 
diameter of 32' 1"-1. At our comet in perihelio the ap- 
parent angular diameter of the sun was no less than 121° 
32'. The ratio of the spherical surfaces thus occupied (as 
appears from spherical geometry) is that of the squares of 
the sines of the fourth parts of these angles to each other, 
or that of 1 : 47042. And in this proportion are to each 
other the amounts of light and heat thrown by the sun on 
an equal area of exposed surface on our earth and at the 
comet in equal instants of time. Let any one imagine 
the effect of so fierce a glare as that of 47000 suns such 
as we experience the warmth of, on the materials of which 
the earth's surface is composed. To form some practical 
idea of it we may compare it with what is recorded of 
Parker's great lens, whose diameter was 32^ inches, and 
focal length six feet eight inches. The effect of this, sup- 
posing all the light and heat transmitted, and the focal 
concentration perfect (both conditions very imperfectly 



502 OUTLINES OF ASTRONOMY 

satisfied), would be to enlarge the sun's effective angular 
diameter to 23° 26', which, compared on the same principle 
with a sun of 32' in diameter, would give a multiplier of 
only 1915, and when increased sevenfold (as was usually the 
case), by interposing a' concentrating lens, 13405 instead of 
47000. The heat to which the comet was subjected there- 
fore surpassed that in the focus of the lens in question, on 
the lowest calculation, in the proportion of 24£ to 1 without, 
or 3.2 to 1 with the concentrating lens. Yet that lens, so 
used, melted carnelian, agate, and rock crystal! 

(593.) To this extremity of heat however the comet was 
exposed but for a short time. Its actual velocity in peri- 
helio was no less than 366 miles per second, and the whole 
of that segment of its orbit above {i.e. north of) the plane of 
the ecliptic, and in which, as will appear from a considera- 
tion of the elements, the perihelion was situated, was de- 
scribed in little more than two hours; such being the whole 
duration of the time from the ascending to the descending 
node, or in which the comet had north latitude. Arrived 
at the descending node, its distance from the sun would be 
already doubled, and the radiation reduced to one-fourth 
of its maximum amount. The comet of 1680, whose peri- 
helion distance was 0'0062, and which therefore approached 
the sun's surface within one-third part of his radius (more 
than double the distance of the comet now in question) was 
computed by Newton to have been subjected to an intensity 
of heat 2000 times that of red-hot iron — a term of comparison 
indeed of a very vague description, and which modern 
thermotics do not recognize as affording a legitimate 
measure of radiant heat. 16 



15 A transit of the comet of 1843 over the sun's disk must probably have 
taken place shortly after its passage through its descending node. It is greatly 



OUTLINES OF ASTRONOMY 508 

(594.) Although, some of the observations of this eomct 
were vague and inaccurate, yet there seem good grounds for 
believing that its whole course cannot be reconciled with 
a parabolic orbit, and that it really describes an ellipse. 
Previous to any calculation, it was remarked that in the 
year 1668 the tail of an immense comet was seen in Lisbon, 
at Bologna, in Brazil, and elsewhere, occupying nearly the 
same situation among the stars, and at the same season of 
the year, viz. on the 5th of March and the following days. 
Its brightness was such that its reflected trace was easily 
distinguished on the sea. The head, when it at length came 
in sight, was comparatively faint and scarce discernible. 
No precise observations were made of this comet, but the 
singular coincidence of situation, season of the year, and 
physical resemblance, excited a strong suspicion of the 
identity of the two bodies, implying a period of 175 years 
within a day or two more or less. This suspicion has been 
converted almost into a certainty by a careful examination 
of what is recorded of the older comet. Locating on a 
celestial chart the situation of the head, concluded from 
the direction and appearance of the tail, when only that 
was seen, and its visible place, when mentioned, according 
to the descriptions given, it has been found practicable to 
derive a rough orbit from the course thus laid down: and 
this agrees in all its features so well with that of the modern 
comet as nearly to remove all doubt on the subject. Comets, 
moreover, are recorded to have been seen in A.D. 268, 442- 



to be regretted that so interesting a phenomenon should have passed unob- 
served. Whether it be possible that some offset of its tail, darted off so Lite 
as the 7th of March, when the comet was already far south of the ecliptic, 
should have crossed that plane and been seen near the Pleiades, may be 
doubted. Certain it is, that on the evening of that day, a decidedly cometic 
ray was seen in the immediate neighborhood of those stars by Mr. Nasmyth. 
(Ast. Soc. Notices, vol. v. p. 270.) 



504 OUTLINES OF ASTRONOMY 

43, 791, 968, 1143, 1317, 1494, which may have been returns 
of this, since the period above mentioned would bring round 
its appearance to the years 268, 443, 618, 793, 968, 1143, 
1318, and 1493, and a certain latitude must always be 
allowed for unknown perturbations. 

(595.) But this is not the only comet on record whose 
identity with the comet of '43 has been maintained. In 
1689 a comet bearing a considerable resemblance to it was 
observed from the 8th to the 23d of December, and from the 
few and rudely observed places recorded, its elements had 
been calculated by Pingre, one of the most diligent inquir- 
ers into this part of astronomy. 17 From these it appears 
that the perihelion distance of that comet was very remark- 
ably small, and a sufficient though indeed rough coinci- 
dence in the places of the perihelion and node tended to 
corroborate the suspicion. Bat the inclination (69°) as- 
signed to it by Pingre appeared conclusive against it. On 
recomputing the elements, however, from his data, Professor 
Pierce has assigned to that comet an inclination widely dif- 
fering from Pingre's, viz. 30° 4', 18 and quite within reason- 
able limits of resemblance. But how does this agree with 
the longer period of 175 years before assigned ? To recon- 
cile this we must suppose that these 175 years comprise at 
least eight returns of the comet, and that in effect a mean 
period of 21 y *875 must be allowed for its return. Now it is 

17 Author of the "Cometographie," a work indispensable to all who would 
study this interesting department of the science. 

18 United States Gazette, May 29, 1843. Considering that all the observa- 
tions lie near the descending node of the orbit, the proximity of the comet at that 
time to the sun, and the loose nature of the recorded observations, no doubt 
almost any given inclination might be deduced from them. The true test in 
such cases is not to ascend from the old incorrect data to elements, but to 
descend from known and certain elements to the older data, and ascertain 
whether the recorded phenomena can be represented by them (perturbations 
included) within fair limits of interpretation. Such is the course pursued 
by Clausen. 



OUTLINES OF ASTRONOMY 505 

worth remarking that this period calculated backward from 
1843-156 will bring us upon a series of years remarkable for 
the appearance of great comets, many of which, as well as 
the imperfect descriptions we have of their appearance and 
situation in the heavens, offer at least no obvious contradic- 
tion to the supposition of their identity with this. Besides 
those already mentioned as indicated by the period of 175 
years, we may specify as probable or possible intermediate 
returns, those of the comets of 1733 ? 19 , 1689 above men- 
tioned, 1559?, 1537, 20 1515, 21 1471, 1426, 1405-6, 1383, 1361, 
1840, 22 1296, 1274, 1230, 23 1208, 1098, 1056, 1034, 1012, 24 
990 ?, 26 925?, 858? ?, 684, 26 552, 530, 27 421, 245 or 247, 28 180 29 , 
158. Should this view of the subject be the true one, we 
may expect its return about the end of 1864 or beginning of 
1865, in which event it will be observable in the Southern 
Hemisphere both before and after its perihelion passage. 30 
(596.) M. Clausen, from the assemblage of all the obser- 



19 P. Passage 1733-781. The great southern comet of May 17th seems too 
early in the year. 

20 P. P. 1536 906. In January 1537, a comet was seen in Pisces. 

21 P. P. 1515-031. A comet predicted the death of Ferdinand the Catholic. 
He died Jan. 23, 1515. 

22 P. P. 1340-031. Evidently a southern comet, and a very probable ap- 
pearance. 

23 P. P. 1230 656, was perhaps a return of Halley's. 

24 P. P. 1011-906. In 1012, a very great comet in the southern part of the 
heavens. "Son eclat blessait les yeux. " (Pingre Cometographie, from whom 
all these recorded appearances are taken.) 

25 P. P. 990-031. "Comete fort epou van table," some year between 989 
and 998. 

26 P. P. 683-781. In 684, appeared two or three comets. Dates begin 
to be obscure. 

27 Two disrinct comets appeared in 530 and 531, the former observed in 
China, the latter in Europe. 

28 P. P. 246*281; both southern comets of the Chinese annals. The year 
of one or other may be wrong. 

29 P. P. 180-656. Nov. 6, A.D. 180. A southern comet of the Chinese 
annals. 

30 Clausen, Astron. Nadir. No. 415. Mr. Cooper's remarks on this comet 
in his Catalogue of Comets (notes, p. xviii.) go to assign by far the greatest 
weight of probability to a period of 35 -ly for this comet. 



506 OUTLINES OF ASTRONOMY 

vations of this comet known to him, has calculated elliptic 
elements which give the extraordinarily short period of 6-38 
years. And in effect it has been suggested that a still fur- 
ther subdivision of the period of 21-875 into three of 7 '292 
years would reconcile this with other remarkable cornels. 
This seems going too far, but at all events the possibility of 
representing its motions by so short an ellipse will easily 
reconcile us to the admission of a period of 21 years. That 
it should only be visible in certain apparitions, and not in 
others, is sufficiently explained by the situation of its orbit. 
(597.) We have been somewhat diffuse on the subject of 
this comet, for the sake of showing the degree and kind of 
interest which attaches to cometic astronomy in the present 
state of the science. In fact, there is no branch of astron- 
omy more replete with interest, and we may add more 
eagerly pursued at present, inasmuch as the hold which 
exact calculation gives us on it may be regarded as com- 
pletely established; so that whatever maybe concluded as 
to the motions of any comet which shall henceforward come 
to be observed, will be concluded on sure grounds and with 
numerical precision; while the improvements which have 
been introduced into the calculation of cometary perturba- 
tion, and the daily increasing familiarity of numerous as- 
tronomers with computations of this nature, enable us to 
trace their past and future history with a certainty, which 
at the commencement of the present century could hardly 
have been looked upon as attainable. Every comet newly 
discovered is at once subjected to the ordeal of a most rigor- 
ous inquiry. Its elements, roughly calculated within a few 
days of its appearance, are gradually approximated to as ob- 
servations accumulate, by a multitude of ardent and expert 
computists. On the least indication of a deviation from a 



OUTLINES OF ASTRONOMY 507 

parabolic orbit, its elliptic elements become a subject of 
universal and lively interest and discussion. Old records 
are ransacked, and old observations reduced, with all the 
advantage of improved data and methods, so as to rescue 
from oblivion the orbits of ancient comets which present 
any similarity to that of the new visitor. The disturbances 
undergone in the interval by the action of the planets are 
investigated, and the past, thus brought into unbroken con- 
nection with the present, is made to afford substantial 
ground for prediction of the future. A great impulse 
meanwhile has been given of late years to the discovery 
of comets by the establishment in 1840, 31 by his late Maj- 
esty the King of Denmark, of a prize medal to be awarded 
for every such discovery, to the first observer (the influence 
of which may be most unequivocally traced in the great 
number of these bodies which every successive year sees 
added to our list), and by the circulation of notices, by spe- 
cial letter, 32 of every such discovery (accompanied, when 
possible, by an ephemeris), to all observers who have shown 
that they take an interest in the inquiry, so as to insure the 
full and complete observation of the new comet so long as 
it remains within the reach of our telescopes. Among the 
observers who have been most successful in the discovery 
of comets, we find no less than 29 discovered by Pons, 14 
by Messier, and 10 by Mechain, 8 by De Yico, 8 by Miss C. 
Herschel — who, however, is not the only female observer of 
these bodies, the comet of 1847 having been independently 
detected by two ladies, Miss Maria Mitchell, of Nantucket, 



31 See the announcement of this institution in Astron. Nadir. No. 400. 
22 By the late Prof. Schumacher, Director of the Royal Observatory of 
Altona. 



Astronomy — Vol. XX — 3 



508 OUTLINES OF ASTRONOMY 

U. S., and Madame Kiimker, of Hamburg, the priority lying 
with the American astronomess. 

(598.) It is by no means merely as a subject of antiqua- 
rian interest, or on account of the brilliant spectacle which 
comets occasionally afford, that astronomers attach a high 
degree of importance to all that regards them. Apart even 
from the singularity and mystery which appertains to their 
physical constitution, they have become, through the me- 
dium of exact calculation, unexpected instruments of in- 
quiry into points connected with the planetary system itself, 
of no small importance. We have seen that the movements 
of the comet Encke, thus minutely and perseveringly traced 
by the eminent astronomer whose name is used to distin- 
guish it, has afforded ground for believing in the presence 
of a resisting medium filling the whole of our system. Simi- 
lar inquiries, prosecuted in the cases of other periodical 
comets, will extend, confirm, or modify our conclusions on 
this head. The perturbations, too, which comets experience 
in passing near any of the planets, may afford, and have 
afforded, information as to the magnitude of the disturbing 
masses, which could not well be otherwise obtained. Thus 
the approach of this comet to the planet Mercury in 1838 
afforded an estimation of the mass of that planet the more 
precious, by reason of the great uncertainty under which all 
previous determinations of that element labored. Its ap- 
proach to the same planet in the year 1848 was still nearer. 
On the 22d of November their mutual distance was only fif- 
teen times the moon's distance from the eartb. 

(599.) It is, however, in a physical point of view that 
these bodies offer the greatest stimulus to our curiosity. 
There is, beyond question, some profound secret and mys- 
tery of nature concerned in the phenomenon of their tails. 



OUTLINES OF ASTRONOMY 609 

Perhaps it is not too much to hope that future observation, 
borrowing every aid from rational speculation, grounded on 
the progress of physical science generally (especially those 
branches of it which relate to the ethereal or imponderable 
elements), may erelong enable us to penetrate this mystery, 
and to declare whether it is really matter in the ordinary ac- 
ceptation of the term which is projected from their heads 
with such extravagant velocity, and if not impelled, at least 
directed in its course by a reference to the sun, as its point 
of avoidance. In no respect is the question as to the mate- 
riality of the tail more forcibly pressed on us for considera- 
tion, than in that of the enormous sweep which it makes 
round the sun in perihelio, in the manner of a straight and 
rigid rod, in defiance of the law of gravitation, nay, even of 
the received laws of motion, extending (as we have seen in 
the comets of 1680 and 1848) from near the sun's surface to 
the earth's orbit, yet whirled round unbroken: in the latter 
case through an angle of 180° in little more than two hours. 
It seems utterly incredible that in such a case it is one and 
the same material object which is thus brandished. If there 
could be conceived such a thing as a negative shadow, a mo- 
mentary impression made upon the iuminiferous ether be- 
hind the comet, this would represent in some degree the 
conception such a phenomenon irresistibly calls up. But 
this is not all. Even such an extraordinary excitement of 
the ether, conceive it as we will, will afford no account of 
the projection of lateral streamers; of the effusion of light 
from the nucleus of a comet toward the sun; and its subse- 
quent ?*ejection ; of the irregular and capricious mode in 
which that effusion has been seen to take place; none of the 
clear indications of alternate evaporation and condensation 
going on in the immense regions of space occupied by the 



510 OUTLINES OF ASTRONOMY 

tail and coma — none, in short, of innumerable other facts 
which link themselves with almost equally irresistible 
cogency to our ordinary notions of matter and force. 

(600.) The great number of comets which appear to move 
in parabolic orbits, or orbits at least indistinguishable from 
parabolas during their description of that comparatively 
small part within the range of their visibility to us, has 
given rise to an impression that they are bodies extraneous 
to our system, wandering through space, and merely yield- 
ing a local and temporary obedience to its laws during their 
sojourn. What truth there may be in this view, we may 
never have satisfactory grounds for deciding. On such a 
hypothesis, our elliptic comets owe their permanent deni- 
zenship within the sphere of the sun's predominant attrac- 
tion to the action of one or other of the planets near which 
they may have passed, in such a manner as to diminish their 
velocity, and render it compatible with elliptic motion. 33 A 
similar cause acting the other way, might with equal proba- 
bility give rise to a hyperbolic motion. But whereas in the 
former case the comet would remain in the system, and 
might make an indefinite number of revolutions, in the lat- 
ter it would return no more. This may possibly be the 
cause of the exceedingly rare occurrence of a hyperbolic 
comet as compared with elliptic ones. 

(601.) All the planets without exception, and almost 
all the satellites, circulate in one direction. Ketrograde 
comets, however, are of very common occurrence, which 
certainly would go to assign them an exterior or at least 
an independent origin. Laplace, from a consideration of 
all the cometary orbits known in the earlier part of the 

33 The velocity in an ellipse is always less than in a parabola, at equal 
distances from the sun; iu a hyperbola always greater. 



OUTLINES OF ASTRONOMY 511 

present century, concluded, that the mean or average situ- 
ation of the planes of all the cometary orbits, with respect 
to the ecliptic, was so nearly that of perpendicularity, as 
to afford no presumption of any cause biasing their direc- 
tions in this respect. Yet we think it worth noticing thah 
among the comets which are as yet known to describe 
elliptic orbits, not one whose inclination is under 17° is 
retrograde; and that out of thirty-six comets which have 
had elliptic elements assigned to them, whether of great or 
small excentricities, and without any limit of inclination, 
only five are retrograde, and of these, only two, viz. Hal- 
ley's and the great comet of 1843, can be regarded as satis- 
factorily made out. Finally, of the 125 comets whose ele- 
ments are given in the collection of Schumacher and Olbers, 
up to 1823, the number of retrograde comets under 10° of 
inclination is only 2 out of 9, and under 20°, 7 out of 23. 84 
A plane of motion, therefore, nearly coincident with the 
ecliptic, and a periodical return, are circumstances emi- 
nently favorable to direct revolution in the cometary as 
they are decisive among the planetary orbits. [Here also 
we may notice a very curious remark of Mr. Hind (Ast. 
Nadir. No. 724), respecting periodic comets, viz. that so 
far as at present known, they divide themselves for the 
most part into two families — the one having periods of 
about 75 years, corresponding to a mean distance about 
that of Uranus; the other corresponding more nearly with 
those of the asteroids, and with a mean distance between 
these small planets and Jupiter. The former group con- 
sists of four members, Halley's comet revolving in 76 years, 
one discovered by Olbers in 74, De Vico's 4th comet in 

34 So in edition of 1850. See, however, Appendix, Table IV., for a more 
recent view of these statistical particulars. 



512 OUTLINES OF ASTRONOMY 

73, and Brorsen's 3d in 75 respectively. Examples of the 
latter group are to be seen in App. Table IV. at the end 
of this volume.] 

We may add, too, a marked tendency in the major axes 
of periodical comets to group themselves about a certain 
determinate direction in space, that is to say, a line point- 
ing to the sphere of the fixed stars northward to 70° long, 
and 30° N. lat. or nearly toward the star A Persei (in the 
Milky Way), and in the southern to a point (also in the 
Milky Way) diametrically opposite. (Ast. Nachr. No. 853.) 

(601 a.) The third great comet of the present century 
(those of 1811 and 1843 being the other two) appeared from 
June 2, 1858, to January, 1859, being known as Donati's 
comet, from its first discoverer. Its head was remarkably 
brilliant; and its tail, like a vast aigrette or gracefully- 
curved plume, extended, when longest, over a space of 
upward of 30°. Its curvature was very marked, deflecting 
toward the region quitted by the comet, as if left behind 
(no proof, as generally supposed, of any resistance experi- 
enced in its motion, but a necessary consequence of the 
combination of its impulse outward from the sun with the 
proper velocity of the comet at the moment of its emission). 
The American observers speak of two long, narrow, per- 
fectly straight rays of faint light, tangents to the limiting 
curves of the aigrette at its quitting the head. The phe- 
nomena of the nucleus under high magnifying powers were 
very complex and remarkable. In each of the years 1861, 
1862, appeared great comets: that of 1861, through, or very 
near whose tail the earth passed on the 30th of June, was 
remarkable for the great length and straightness of one side 
of its tail; that of 1862 for the high condensation of the 
nucleus and of the single jet issuing from it. 



OUTLINES OF ASTRONOMY 51o 



PART II 

OF THE LUNAR AND PLANETARY PERTURBATIONS 
"Magnus ab integro sasclorum nascitur ordo." — Yirg. Pollio. 

CHAPTEB XII 

Subject Propounded — Problem of Three Bodies — Superposition of Small 
Motions — Estimation of the Disturbing Force — Its Geometrical Rep- 
resentation — Numerical Estimation in Particular Cases — Resolution 
into Rectangular Components — Radial, Transversal and Orthogonal 
Disturbing Forces — Normal and Tangential — Their Characteristic Ef- 
fects — Effects of the Orthogonal Force — Motion of the Nodes — Con- 
ditions of their Advance and Recess — Cases of an Exterior Planet 
Disturbed by an Interior — The Reverse Case — In Every Case the 
Node of the Disturbed Orbit Recedes on the Plane of the Disturb- 
ing on an Average — Combined Effect of Many Such Disturbances — 
Motion of the Moon's Nodes — Change of Inclination — Conditions of 
its Increase and Diminution — Average Effect in a Whole Revolution 
— Compensation in a Complete Revolution of the Nodes — Lagrange's 
Theorem of the Stability of the Inclinations of the Planetary Orbits 
— Change of Obliquity of the Ecliptic — Precession of the Equinoxes 
Explained — Nutation — Principle of Forced Vibrations 

(602.) In the progress of this work, we have more than 
once called the reader's attention to the existence of in- 
equalities in the lunar and planetary motions not included 
in the expression of Kepler's laws, but in some sort supple- 
mentary to them, and of an order so far subordinate to those 
leading features of the celestial movements, as to require, 
for their detection, nicer observations, and longer-continued 
comparison between facts and theories, than suffice for the 
establishment and verification of the elliptic theory. These 
inequalities are known, in physical astronomy, by the name 
of perturbations. They arise, in the case of the primary 



514 OUTLINES OF ASTRONOMY 

planets, from the mutual gravitations of these planets to- 
ward each other, which derange their elliptic motions round 
the sun; and in that of the secondaries, partly from the 
mutual gravitation of the secondaries of the same system 
similarly deranging their elliptic motions round their com- 
mon primary, and partly from the unequal attraction of 
the sim and planets on them and on their primary. These 
perturbations, although small, and, in most instances, in-, 
sensible in short intervals of time, yet when accumulated, 
as some of them may become, in the lapse of ages, alter 
very greatly the original elliptic relations, so as to render 
the same elements of the planetary orbits, which at one 
epoch represented perfectly well their movements, inade- 
quate and unsatisfactory after long intervals of time. 

(603.) When New ton first reasoned his way from the 
broad features of the celestial motions, up to the law of 
universal gravitation, as affecting all matter, and rendering 
every particle in the universe subject to the influence of 
every other, he was not unaware of the modifications which 
this generalization would induce upon the results of a more 
partial and limited application of the same law to the revo- 
lutions of the planets about the sun, and the satellites about 
their primaries, as their only centres of attraction. So far 
from it, his extraordinary sagacity enabled him to perceive 
very distinctly how several of the most important of the 
lunar inequalities take their origin, in this more general 
way of conceiving the agency of the attractive power, espe- 
cially the retrograde motion of the nodes, and the direct 
revolution of the apsides of her orbit. And if he did not 
extend his investigations to the mutual perturbations of the 
planets, it was not for want of perceiving that such pertur- 
bations must exist, and might go the length of producing 



OUTLINES OF ASTRONOMY 515 

great derangements from the actual state of the system, but 
was owing to the then undeveloped state of the practical 
part of astronomy, which had not yet attained the precision 
requisite to make such an attempt inviting, or indeed feasi- 
ble. What Newton left undone, however, his successors 
have accomplished; and, at this day, it is hardly too much 
to assert that there is not a single perturbation, great or 
small, which observation has become precise enough clearly 
to detect and place in evidence which has not been traced 
up to its origin in the mutual gravitation of the parts 
of our system, and minutely accounted for, in its numeri- 
cal amount and value, by strict calculation on Newton's 
principles. 

(60-i.) Calculations of this nature require a very high 
analysis for their successful performance, such as is far 
beyond the scope and object of this work to attempt ex- 
hibiting. The reader who would master them must prepare 
himself for the undertaking by an extensive course of pre- 
paratory study, and must ascend by steps which we must 
not here even digress to point out. It will be our object, 
in this chapter, however, to give some general insight into 
the nature and manner of operation of the acting forces, 
and to point out what are the circumstances which, in some 
cases, give them a high degree of efficiency — a sort of pur- 
chase on the balance of the system; while, in others, with 
no less amount of intensity, their effective agency in pro- 
ducing extensive and lasting changes is compensated or 
rendered abortive; as well as to explain the nature of 
those admirable results respecting the stability of our sys- 
tem, to which the researches of geometers have conducted 
them; and which, under the form of mathematical theorems 
of great simplicity and elegance, involve the history of the 



526 OUTLINES 0$ ASTRONOMY 

past and future state of the planetary orbits during ages 
of which, contemplating the subject in this point of view, 
we neither perceive the beginning nor the end. 

(605.) Were there no other bodies in the universe but 
the sun and one planet, the latter would describe an exact 
ellipse about the former (or both round their common cen- 
tre of gravity), and continue to perform its revolutions in 
one and the same orbit forever; but the moment we add 
to our combination a third body, the attraction of this will 
draw both the former bodies out of their mutual orbits, 
and, by acting on them unequally, will disturb their rela- 
tion to each other, and put an end to the rigorous and 
mathematical exactness of their elliptic motions, not only 
about a fixed point in space, but about one another. From 
this way of propounding the subject, we see that it is not 
the whole attraction of the newly introduced body which 
produces perturbation, but the difference of its attractions 
on the two originally present. 

(606.) Compared to the sun, all the planets are of ex- 
treme minuteness; the mass of Jupiter, the greatest of them 
all, being not more than about one 1100th part that of the 
sun. Their attractions on each other, therefore, are all 
very feeble, compared with the presiding central power, 
and the effects of their disturbing forces are proportionally 
minute. In the case of the secondaries, the chief agent by 
which their motions are deranged is the sun itself, whose 
mass is indeed great, but whose disturbing influence is im- 
mensely diminished by their near proximity to their prima- 
ries, compared to their distances from the sun, which ren- 
ders the difference of attractions on both extremely small, 
compared to the whole amount. In this case the greatest 
part of the sun's attraction, viz. that which is common to 



OUTLINES OF ASTRONOMY 517 

both, is exerted to retain both primary and secondary in 
their common orbit about itself, and prevent their parting 
company. Only the small overplus of force on one as com- 
pared with the other acts as a disturbing power. The 
mean value of this overplus, in the case of the moon dis- 
turbed by the sun, is calculated by Newton to amount to 
no higher a fraction than 6 J m of gravity at the earth's sur- 
face, or ,} 9 of the principal force which retains the moon in 
its orbit. 

(607.) From this extreme minuteness of the intensities 
of the disturbing, compared to the principal forces, and the 
consequent smallness of their momentary effects, it happens 
that we can estimate each of these effects separately, as if 
the others did not take place, without fear of inducing error 
in our conclusions beyond the limits necessarily incident 
to a first approximation. It is a principle in mechanics, 
immediately flowing from the primary relations between 
forces and the motions they produce, that when a number of 
very minute forces act at once on a system, their joint effect 
is the sum or aggregate of their separate effects, at least 
within such limits, that the original relation of the parts of 
the system shall not have been materially changed by their 
action. Such effects supervening on the greater movements 
due to the action of the primary forces may be compared to 
the small ripplings caused by a thousand varying breezes 
on the broad and regular swell of a deep and rolling ocean, 
which run on as if the surface were a plane, and cross in 
all directions without interfering, each as if the other had 
no existence. It is only when their effects become accu- 
mulated in lapse of time, so as to alter the primary rela- 
tions or data of the system, that it becomes necessary to 
have especial regard to the changes correspondingly intro- 



518 OUTLINES OF ASTRONOMY 

duced into the estimation of their momentary efficiency, by 
which the rate of the subsequent changes is affected, and 
periods or cycles of immense length take their origin. From 
this consideration arise some of the most curious theories ot 
physical astronomy. 

(60S.) Hence it is evident, that in estimating the disturb- 
ing influence of several bodies forming a system, in which 
one has a remarkable preponderance over all the rest, we 
need not embarrass ourselves with combinations of the dis- 
turbing powers one among another, unless where immensely 
long periods are concerned; such as consist of many hun- 
dreds of revolutions of the bodies in question about their 
common centre. So that, in effect, so far as we propose to 
go into its consideration, the problem of the investigation 
of the perturbations of a system, however numerous, consti- 
tuted as ours is, reduces itself to that of a system of three 
bodies: a predominant central body, a disturbing, and a 
disturbed; the two latter of which may exchange denomi- 
nations, according as the motions of the one or the other 
are the subject of inquiry. 

(609.) Both the intensity and direction of the disturb- 
ing force are continually varying, according to the relative 
situation of the disturbing and disturbed body with respect 
to the sun. If the attraction of the disturbing body M, on 
the central body S, and the disturbed body P (by which 
designations, for brevity, we shall hereafter indicate them), 
were equal, and acted in parallel lines, whatever might 
otherwise be its law of variation, there would be no devia- 
tion caused in the elliptic motion of P about S, or of each 
about the other. The case would be strictly that of art. 454; 
the attraction of M, so circumstanced, being at every mo- 
ment exactly analogous in its effects to terrestrial gravity, 



OUTLINES OF ASTRONOMY 519 

which acts in parallel lines, and is equally intense on all 
bodies, great and small. But this is not the case of nature. 
Whatever is stated in the subsequent article to that last 
cited, of the disturbing effect of the sun and moon, is, 
mutatis mutandis, applicable to every case of perturbation; 
and it must be now our business to enter, somewhat more 
in detail, into the general heads of the subject there merely 
hinted at. 

(610.) To obtain clear ideas of the manner in which the 
disturbing force produces its various effects, we must ascer- 
tain at any given moment, and in any relative situations of 
the three bodies, its direction and intensity as compared 
with the gravitation of P toward S, in virtue of which 
latter force alone P would describe an ellipse about S re- 
garded as fixed, or rather P and S about their common 
centre of gravity in virtue of their mutual gravitation to 
each other. In the treatment of the problem of three 
bodies, it is convenient, and tends to clearness of appre- 
hension, to regard one of them as fixed, and refer the mo- 
tions of the others to it as to a relative centre. In the case 
of two planets disturbing each other's motions, the sun is 
naturally chosen as this fixed centre; but in that of satel- 
lites disturbing each other, or disturbed by the sun, the 
centre of their primary is taken as their point of reference, 
and the sun itself is regarded in the light of a very distant 
and massive satellite revolving about the primary in a 
relative orbit, equal and similar to that which the primary 
describes absolutely round the sun. Thus the generality of 
our language is preserved, and when, referring to any par- 
ticular central body, we speak of an exterior and an interior 
planet, we include the cases in which the former is the sun 
and the latter a satellite; as, for example, in the Lunar 



520 OUTLINES OF ASTRONOMY 

theory. It is a principle in dynamics, that the relative 
motions of a system of bodies inter se are no way altered 
by impressing on all of them a common motion or motions, 
or a common force or forces accelerating or retarding them 
all equally in common directions, i.e. in parallel lines. 
Suppose, therefore, we apply to all the three bodies, S, P, 
and M, alike, forces equal to those with which M and P 
attract S, but in opposite directions. Then will the relative 
motions both of M and P about S be unaltered; but S, being 
now urged by equal and opposite forces to and from both 
M and P, will remain at rest. Let us now consider how 
either of the other bodies, as P, stands affected by these 
newly introduced forces, in addition to those which before 
acted on it. It is clear that now P will be simultaneously 
acted on by four forces; first, the attraction 0/ S in the 
direction P S; secondly, an additional force, in the same 
direction, equal to its attraction on S; thirdly, the attraction 
ofM. in the direction P M; and fourthly, a force parallel to 
M S, and equal to M's attraction on S. Of these, the first 
two following the same law of the inverse square of the 
distance S P, may be regarded as one force, precisely as if 
the sum of the masses of S and P were collected in S ; and 
in virtue of their joint action, P will describe an ellipse 
about S, except in so far as that elliptic motion is disturbed 
by the other two forces. Thus we see that in this view of 
the subject the relative disturbing force acting on P is no 
longer the mere single attraction of M, but a force resulting 
from the composition of that attraction with M's attraction 
on S transferred to P in a contrary direction. 

(611.) Let P A be part of the relative orbit of the 
disturbed, and M B of the disturbing body, their planes 
intersecting in the line of nodes SAB, and having to 



OUTLINES OF ASTRONOMY 



621 



each other the inclination expressed by the spherical angle 
P A a. In M P, produced if required, take M N : MS:: 
M S a : M P 3 . Then, if S M 1 be taken to represent, in 
quantity and direction, the accelerative attraction of M on S, 
M S will represent in quantity and direction the new force 
applied to P, parallel to that line, and N M will represent 
on the same scale the accelerative attraction of M on P. 
Consequently, the disturbing force acting on P will be the 




resultant of two forces applied at P, represented respectively 
by N M and M S, which by the laws of dynamics are 
equivalent to a single force represented in quantity and 
direction by K S, but having P for its point of application. 
(612.) The line N S is easily calculated by trigonom- 
etry, when the relative situations and real distances of the 
bodies are known; and the force expressed by that line is 
directly comparable with the attractive forces of S on P by 



1 The reader will be careful to observe the order of the letters, where forces 
are represented by lines. M S represents a force acting from M toward S, S M 
from S toward M. 



522 



OUTLINES OF ASTRONOMY 



the following proportions, in which M, S, represent the 
masses of those bodies which are supposed to be known, 
and to which, at equal distances, their attractions are 
proportional: 

Disturbing force : M's attraction on S :: N S : S M; 

M's attraction on S : S's attraction on M :: M : S; 

S's attraction on M : S's attraction on P :: S P 2 : S M a ; 
by compounding which proportions we collect as follows: 

Disturbing force : S's attraction on P :: M . N S . S P a : 
S . S M 3 . 

A few numerical examples are subjoined, exhibiting the 
results of this calculation in particular cases, chosen so as 
to exemplify its application under very various circum- 
stances, throughout the planetary system. In each case 
the numbers set down express the proportion in which the 
central force retaining the disturbed body in its elliptic 
orbit exceeds the disturbing force, to the nearest whole 
number. The calculation is made for three positions of the 
disturbing body, viz, at its greatest, its least, and its mean 
distance from the disturbed. 



Disturbing Body. 


Disturbed Body. 


Ratio at the 
greatest Dis- 
tance : 1. 


Ratio at the 

mean Distance 

:1. 


Ratio at the 

least Distance 

:1. 


The Sun - - 


The Moon - - 


90 


179 


89 


Jupiter - - - 


Saturn - - - 


354 


312 


128 


Jupiter - - - 


The Earth - - 


95683 


147575 


53268 


Venus - - 


The Earth 


255208 


210245 


26833 


Neptune - - 


Uranus - - - 


57420 


56592 


5519 


Mercury - - 
Jupiter - - - 


Neptune - - 
Ceres 


845 
6433 


845 
6937 


845 
1033 


Saturn - - - 


Jupiter - - - 


20248 


21579 


3065 



(613.) If the orbit of the disturbing body be circular, 
S M is invariable. In this case, N S will continue to rep- 
resent the disturbing force on the same invariable scale, 



OUTLINES OF ASTRONOMY 



523 



whatever may be the configuration of the three bodies with 
respect to each other. If the orbit of M be but little elliptic, 
the same will be nearly the case. In what follows through- 
out this chapter, except where the contrary is expressly 
mentioned, we shall neglect the excentricity of the disturb- 
ing orbit. 

(614.) If P be nearer to M than S is, M K is greater than 
M P, and N lies in M P prolonged, and therefore on the 
opposite side of the plane of P's orbit from that on which 
M is situated. The force N S therefore urges P toward M's 
plane, and toward a point X, situated between S and M, in 
the line S M. If ^he distance M P be equal to M S as when 
P is situated, suppose at D or E, M N is also equal to M P 




or M S, so that N coincides with P, and therefore X with 
S, the disturbing forces being in these cases directed toward 
the central body. But if M P be greater than MS,MN 
is less than M P, and N" lies between M and P, or on the 
same side of the plane of P's orbit that M is situated on. 
The force N S, therefore, applied at P, urges P toward the 
contrary side of that plane toward a point in the line M S 
produced, so that X now shifts to the further side of S. In 



524 OUTLINES OF ASTRONOMY 

all cases, the disturbing force is wholly effective in the 
plane M P S, in which the three bodies lie. 

It is very important for the student to fix distinctly and 
bear constantly in his mind these relations of the disturbing 
agency considered as a single unresolved force, since their 
recollection will preserve him from any mistakes in conceiv- 
ing the mutual actions of the planets, etc., on each other. 
For example, in the figures here referred to, that of art. 
611, corresponds to the case of a nearer disturbed by a more 
distant body, as the earth by Jupiter, or the moon by the 
Sun; and that of the present article to the converse case: 
as, for instance, of Mars disturbed by the earth. Now, in 
this latter class of cases, whenever M P is greater than M S, 
or S P greater than 2 S M, N lies on the same side of the 
plane of P's orbit with M, so that N S, the disturbing force, 
contrary to what might at first be supposed, always urges 
the disturbed planet out of the plane of its orbit toward the 
opposite side to that on which the disturbing planet lies. It 
will tend greatly to give clearness and definiteness to his 
ideas on the subject, if he will trace out on various supposi- 
tions as to the relative magnitude of the disturbing and dis- 
turbed orbits (supposed to lie in one plane) the form of the 
oval about M considered as a fixed point, in which the point 
N lies when P makes a complete revolution round S. 

(615.) Although it is necessary for obtaining in the first 
instance a clear conception of the action of the disturbing 
force, to consider it in this way as a single force having a 
definite direction in space and a determinate intensity, yet 
as that direction is continually varying with the position of 
N S, both with respect to the radii S P, S M, the distance 
P M, and the direction of P's motion, it would be impos- 
sible, by so considering it, to attain clear views of its dy- 



OUTLINES OF ASTRONOMY 525 

namical effect after any considerable lapse of time, and it 
therefore becomes necessary to resolve it into other equiva- 
lent forces acting in such directions as shall admit of distinct 
and separate consideration. Now this may be done in sev- 
eral different modes. First, we may resolve it into three 
forces acting in fixed directions in space rectangular to one 
another, and by estimating its effect in each of these three 
directions separately, conclude the total or joint effect. 
This is the mode of procedure which affords the readiest 
and most advantageous handle to the problem of perturba- 
tions when taken up in all its generality, and is accordingly 
that resorted to by geometers of the modern school in all 
their profound researches on the subject. Another mode 
consists in resolving it also into three rectangular compo- 
nents, not, however, in fixed directions, but in variable ones, 
viz. in the directions of the lines 1ST Q, Q L, and L S, of 
which L S is in the direction of the radius vector S P, Q L 
in a direction perpendicular to it, and in the plane in which 
S P and a tangent to P's orbit at P both lie; and lastly, 
1ST Q in a direction perpendicular to the plane in which P is 
at the instant moving about S. The first of these resolved 
portions we may term the radial component of the disturb- 
ing force, or simply the radial disturbing force; the second 
the transversal', and the third the orthogonal* When the 
disturbed orbit is one of small excentricit^ the transversal 
component acts nearly in the direction of the tangent to P's 
orbit at P, and is therefore confounded with that resolved 
component which we shall presently describe (art. 618) 
under the name of the tangential force. This is the mode 



2 This is a term coined for the occasion. The want of some appellation for 
this component of the disturbing force is often felt. 



526 OUTLINES OF ASTRONOMY 

of resolving the disturbing force followed by Newton and 
his immediate successors,, 

(616.) The immediate actions of these components of the 
disturbing force are evidently independent of each other, 
being rectangular in their directions; and they affect the 
movement of the disturbed body in modes perfectly distinct 
and characteristic. Thus, the radial component, being 
directed to or from the central body, has no tendency to 
disturb either the plane of P's orbit, or the equable descrip- 
tion of areas by P about S, since the law of areas propor- 
tional to the times is not a character of the force of gravity 
only, but holds good equally, whatever be the force which 
retains a body in an orbit, provided only its direction is al- 
ways toward a fixed centre. 3 Inasmuch, however, as its law 
of variation is not conformable to the simple law of gravity, 
it alters the elliptic form of P's orbit, by directly affecting 
both its curvature and velocity at every point. In virtue, 
therefore, of the action of this disturbing force, the orbit 
deviates from the elliptic form by the approach or recess of 
P to or from S, so that the effect of the perturbations pro- 
duced by this part of the disturbing force falls wholly on 
the radius vector of the disturbed orbit. 

(617.) The transversal disturbing force represented by 
Q L, on the other hand, has no direct action to draw P to 
or from S. Its whole efficiency is directed to accelerate or 
retard P's motioD in a direction at right angles to S P. 
Now the area momentarily described by P about S, is, 
cceteris paribus, directly as the velocity of P in a direction 
perpendicular to S P. Whatever force, therefore, increases 
this transverse velocity of P, accelerates the description of 

3 Newton, i. 1. 



OUTLINES OF ASTRONOMY 527 

areas, and vice versa. With the area A S P is directly con- 
nected, by the nature of the ellipse, the angle ASP de- 
scribed or to be described by P from a fixed line in the 
plane of the orbit, so that any change in the rate of descrip- 
tion of areas ultimately resolves itself into a change in the 
amount of angular motion about S, and gives rise to a de- 
parture from the elliptic laws. Hence arise what are called 
in the perturbational theory equations (i.e. changes or fluc- 
tuations to and fro about an average quantity) of the mean 
motion of the disturbed body. 

(618.) There is yet another mode of resolving the dis- 
turbing force into rectangular components, which, though 
not so well adapted to the computation of results, in reduc- 
ing to numerical calculation the motions of the disturbed 
body, is fitted to afford a clearer insight into the nature of 
the modifications which the form, magnitude, and situation 
of its orbit undergo in virtue of its action, and which we 
shall therefore employ in preference. It consists in estimat- 
ing the components of the disturbing force, which lie in the 
plane of the orbit, not in the direction we have termed 
radial and transversal, i.e. in that of the radius vector P S 
and perpendicular to it, but in the direction of a tangent to 
the orbit at P, and in that of a normal to the carve, and at 
right angles to the tangent, for which reason these compo- 
nents may be called the tangential and normal disturbing 
forces. When the orbit of the disturbed body is circular, 
or nearly so, this mode of resolution coincides with or differs 
but little from the former, but when the ellipticity is con- 
siderable, these directions may deviate from the radial and 
transversal directions to any extent. As, in the Newtonian 
mode of resolution, the effect of the one component falls 
wholly upon the approach and recess of the body P to the 



528 OUTLINES OF ASTRONOMY 

central body S, and of the other wholly on the rate of de- 
scription of areas by P round S, so in this which we are now 
considering, the direct effect of the one component (the nor- 
mal) falls wholly on the curvature cf the orbit at the point 
of its action, increasing that curvature when the normal 
force acts inward, or toward the concavity of the orbit, and 
diminishing it when in the opposite direction; while, on the 
other hand, the tangential component is directly effective on 
the velocity of the disturbed body, increasing or diminish- 
ing it according as its direction conspires with or opposes its 
motion. It is evident enough that where the object is to 
trace simply the changes produced by the disturbing force, 
in angle and distance from the central body, the former mode 
of resolution must have the advantage in perspicuity of 
view and applicability to calculation. It is less obvious, 
but will abundantly appear in the sequel that the latter 
offers peculiar advantages in exhibiting to the eye and the 
reason the momentary influence of the disturbing force on 
the elements of the orbit itself. 

(619.) Neither of the last-mentioned pairs of resolved 
portions of the disturbing force tends to draw P out of the 
plane of its orbit PSA. But the remaining or orthogonal 
portion N Q acts directly and solely to produce that effect. 
In consequence, under the influence of this force, P must 
quit that plane, and (the same cause continuing in action) 
must describe a curve of double curvature as it is called, no 
two consecutive portions of which lie in the same plane 
passing through S. The effect of this is to produce a con- 
tinual variation in those elements of the orbit of P on which 
the situation of its 'plane in space depends, i.e. on its inclina- 
tion to a fixed plane, and the position in such a plane of the 
node or line of its intersection therewith. As this, among 



OUTLINES OF ASTRONOMY 529 

all the various effects of perturbation, is that which is at 
once the most simple in its conception, and the easiest to 
follow into its remoter consequences, we shall begin with its 
explanation. 

(620.) Suppose that up to P (arts. 611, 614) the body were 
describing an undisturbed orbit C P. Then at P it would 
be moving in the direction of a tangent P K to the ellipse 
P A, which prolonged will intersect the plane of M's orbit 
somewhere in the line of nodes, as at E. Now, at P, let the 
disturbing force parallel to 1ST Q act momentarily on P; then 
P will be deflected in the direction of that force, and instead 
of the arc P p, which it would have described in the next 
instant if undisturbed, will describe the arc P q lying in the 
state of things represented in art. 611 below, and in art. 
611 above, P p with reference to the plane PSA. Thus, 
by this action of the disturbing force, the plane of P's orbit 
will have shifted its position in space from P Sp (an ele- 
mentary portion of the old orbit) to P S q, one of the new. 
Now the line of nodes S A B in the former is determined 
by prolonging P p into the tangent P K, intersecting the 
plane M S B in E, and joining S E. And in like manner, 
if we prolong P q into the tangent P r, meeting the same 
plane in r, and join S r, this will be the new line of nodes. 
Thus we see that, under the circumstances expressed in the 
former figure, the momentary action of the orthogonal dis- 
turbing force will have caused the line of nodes to retrograde 
upon the plane of the orbit of the disturbing body, and 
under those represented in the latter to advance. And it is 
evident that the action of the other resolved portions of the 
disturbing force will not in the least interfere with this re- 
sult, for neither of them tends either to carry P out of its 
former plane of motion, or to prevent its quitting it. Their 



530 OUTLINES OF ASTRONOMY 

influence would merely go to transfer the points of intersec- 
tion of the tangents P p or P q from E or r to B/ or r', points 
nearer to or farther from S than R r, but in the same lines. 

(621.) Supposing, now, M to lie to the left instead of the 
right side of the line of nodes in^. 1, P retaining its situa- 
tion, and M P being less than M S, so that X shall still lie 
between M and S. In this situation of things (or configura- 
tion, as it is termed of the three bodies with respect to each 
other), 1ST will lie below the plane ASP, and the disturbing 
force will tend to raise the body P above the plane, the re- 
solved orthogonal portion N Q in this case acting upward. 
The disturbed arc Fq will therefore lie above P^>, and 
when prolonged to meet the plane MSB, will intersect it 
in a point in advance of R; so that in this configuration the 
node will advance upon the plane of the orbit of M, pro- 
vided always that the latter orbit remains fixed, or, at least, 
does not itself shift its position in such a direction as to de- 
feat this result. 

(622.) Generally speaking, the node of the disturbed orbit 
will recede upon any plane which we may consider as fixed 1 
whenever the action of the orthogonal disturbing force tends 
to bring the disturbed body nearer to that plane; and vice 




versa. This will be evident on a mere inspection of the an- 
nexed figure, in which C A represents a semicircle of the 
projection of the fixed plane as seen from S on the sphere of 
the heavens, and C P A that of the plane of P's undisturbed 
orbit, the motion of P being in the direction of the arrow, 
from C the ascending, to A the descending node. It is at 



OUTLINES OF ASTRONOMY 531 

once seen, by prolonging P q, P q' into arcs of great circles, 
P r, P r' (forward or backward, as the case may be) to meet 
C A, that the node will have retrograded through the arc 
A r, or C r, whenever P q lies between C P A and C A, or 

when the perturbing force carries P toward the fixed plane, 
but will have advanced through A r' or C r' when P q' lies 
above C P A, or when the disturbing impulse has lifted P 
above its old orbit or away from the fixed plane, and this 
without any reference to whether the undisturbed orbitual mo- 
tion ofV at the moment is carrying it toward the plane C A or 
from it, as in the two cases represented in the figure. 

(623.) Let us now consider the mutual disturbance of two 
bodies M and P, in the various configurations in which they 
may be presented to each other and to their common central 
body. And first, let us take the case, as the simplest, where 
the disturbed orbit is exterior to that of the disturbing body 
(as in fig. art. 614), and the distance between the orbits 
greater than the semiaxis of the smaller. First, let both 
planets lie on the same side of the line of nodes. Then (as 
in art. 620) the direction of the whole disturbing force, and 
therefore also that of its orthogonal component, will be 
toward the opposite side of the plane of P's orbit from that 
on which M lies. Its effect therefore will be to draw P out 
of its plane in a direction from the plane of M's orbit, so 
that in this state of things the node will advance on the lat- 
ter plane, however P. and M may be situated in these semi- 
circumferences of their respective orbits. Suppose, next, M 
transferred to the opposite side of the line of nodes, then 
will the direction of its action on P, with respect to the plane 
of P's orbit, be reversed, and P in quitting that plane will 
now approach to instead of receding from the plane of M's 

orbit, so that its node will now recede on that plane. 

Astronomy — Vol. XX — 4 



532 OUTLINES OF ASTRONOMY 

(624.) Thus, while M and P revolve about S, and in the 
course of many revolutions of each are presented to each 
other and to S in all possible configurations, the node of P's 
orbit will always advance on M's when both bodies are on 
the same side of the line of nodes, and recede when on the 
opposite. They will therefore, on an average, advance and 
recede during equal times (supposing the orbits nearly cir- 
cular). And, therefore, if their advance were at each in- 
stant of its duration equally rapid with their recess at each 
corresponding instant during that phase of the movement, 
they would merely oscillate to and fro about a mean posi- 




tion, without any permanent motion in either direction. 
But this is not the case. The rapidity of their recess in 
every position favorable to recess is greater than that of 
their advance in the corresponding opposite position. To 
show this, let us consider any two configurations in which 
M's phases are diametrically opposite, so that the triangles 
P S M, P S M', shall lie in one plane, having any inclination 
to P's orbit, according to the situation of P. Produce P S, 
and draw M m W m' perpendicular to it, which will therefore 
be equal. Take M N : M S : : M S 2 : M P 2 , and M' N' : 
M' S : : M' S 2 : M' P 2 : then, if the orbits be nearly circles, 
and therefore MS = M' S, W M' will be less than M N ; and 



OUTLINES OF ASTRONOMY 533 

therefore (since P M' is greater than P M) P N' : P M' in a 
greater ratio than P N : P M; and consequently, by similar 
triangles, drawing N n, N' n' perpendicular to P S, N' n' : 
if //?' in a greater ratio than N n : M wi, and therefore N' n 
is greater than N n. Now the plane P M M' intersects P's 
orbit in P S, and being inclined to that orbit at the same 
angle through its whole extent, if from N and W perpen- 
diculars be conceived let fall on that orbit, these will be to 
each other in the proportion of N n, N' n' \ and therefore 
the perpendicular from N' will be greater than that from N. 
Now since by art. 611 N' S and N S represent in quantity 
and direction the total disturbing forces of M' and M on P 
respectively, therefore these perpendiculars express (art. 615) 
the orthogonal disturbing forces, the former of which tends 
(as above shown) to make the nodes recede, and the latter to 
advance; and therefore the preponderance in every such 
pair of situations of M is in favor of a retrograde motion. 

(625.) Let us next consider the case where the distance 
between the orbits is less than the semiaxis of the interior, 
or in which the least distance of M from P is less than M S. 
Take any situation of P with respect to the line of nodes 




A C. Then two points d and e, distant by less than 120°, 
can be taken on the orbit of M equidistant from P with S. 
Suppose M to occupy successively every possible situation 
in its orbit, P retaining its place; — then, if it were not for 
the existence of the arc d e, in which the relations of art. 624 



534 OUTLINES OF ASTRONOMY 

are reversed, it would appear by the reasoning of that article 
that the motion of the node is direct when M occupies any 
part of the semiorbit F M B, and retrograde when it is in 
the opposite, but that the retrograde motion on the whole 
would predominate. Much more then will it predominate 
when there exists an arc d M e within which if M be placed, 
its action will produce a retrograde instead of a direct 
motion. 

(626.) This supposes that the arc d e lies wholly in the 
semicircle FdB. But suppose it to lie, as in the annexed 
figure, partly within and partly without that circle. The 
greater part d B necessarily lies within it, and not only so, 
but within that portion, the point of M's orbit nearest to P, 
in which, therefore, the retrograding force has its maximum, 




is situated. Although, therefore, in the portion B e, it is 
true, the retrograde tendency otherwise general over the 
whole of that semicircle (art. 624) will be reversed, yet the 
effect of this will be much more than counterbalanced by 
the more energetic and more prolonged retrograde action 
over d B; and, therefore, in this case also, on the average 
of every possible situation of M, the motion of the node will 
be retrograde. 

(627.) Let us lastly consider an interior planet disturbed 
by an exterior. Take M D and M E {fig. of art. 611), each 
equal to M S. Then first, when P is between D and the 
node A, being nearer than S to M, the disturbing force acts 



OUTLINES OF ASTRONOMY 535 

toward M's orbit on the side on which M lies, and the node 
recedes. It also recedes when (M retaining the same situa- 
tion) P is in any part of the arc E C from E to the other 
node, because in that situation the direction of the disturb- 
ing force, it is true, is reversed, but that portion of P's orbit 
being also reversely situated with respect to the plane of 
M's, P is still urged toward the latter plane, but on the side 
opposite to M. Thus (M holding its place), whenever P is 
anywhere in D A or EC, the node recedes. On the other 
hand, it advances whenever P is between A and E or be- 
tween C and D, because, in these arcs, only one of the two 
determining elements (viz. the direction of the disturbing 
force with respect to the plane of P's orbit; and the situa- 
tion of the one plane with respect to the other as to above 
and below) has undergone reversal. Now first, whenever 
M is anywhere but in the line of nodes, the sum of the arcs 
D A and E C exceeds a semicircle, and that the more, the 
nearer M is to a position at right angles to the line of nodes. 
Secondly, the arcs favorable to the recess of the node com- 
prehend those situations in which the orthogonal disturbing 
force is most powerful, and vice versa. This is evident, be- 
cause as P approaches D or E, this con ponent decreases, 
and vanishes at those points (612). The movement of the 
node itself also vanishes when P comes to the node, for al- 
though in this position the disturbing orthogonal force 
neither vanishes nor changes its direction, yet, since at the 
instant of P's passing the node (A) the recess of the node is 
changed into an advance, it must necessarily at that point 
be stationary. 4 Owing to both these causes, therefore (that 



4 It would seem, at first sight, as if a change per saltum took place here, 
but the continuity of the node's motion will be apparent ' from an inspection of 
tha annexed figure, where b a d 13 a portion of P's disturbed path near tho 



536 



OUTLINES OF ASTRONOMY 



the node recedes during a longer time than it advances, and 
that a more energetic force acting in its recess causes it to 
recede more rapidly), the retrograde motion will preponder- 
ate on the whole in each complete synodic revolution of P. 
And it is evident that the reasoning of this and the fore- 
going articles is no way vitiated by a moderate amount 
of excentricity in either orbit. 

(628.) It is therefore a general proposition, that on the 
average of each complete synodic revolution, the node of 




every disturbed planet recedes upon the orbit of the disturb- 
ing one, or, in other words, that in every pair of orbits, the 
node of each recedes upon the other, and of course upon any 
intermediate plane which we may regard as fixed. On a 
plane not intermediate between them, however, the node of 
one orbit will advance, and that of the other will recede. 

node A, concave toward the plane G- A. The momentary place of the moving 
node is determined by the intersection of the tangent b e with A G-, which as 




I passes through a to d, recedes from A to a, rests there for an instant, and 
then advances again. 



OUTLINES OF ASTRONOMY 537 

Suppose, for instance, C A C to be a plane intermediate be- 
tween P P and M M the two orbits. If p p and m m be the 
new positions of the orbits, the node of P on M will have 
receded from A to 5, that of M on P from A to 4, that of P 
and M on C C respectively from A to 1 and from A to 2. 
But if F A F be a plane not intermediate, the node of M on 
that plane has receded from A to 6, but that of P will have 
advanced from A to 7. If the fixed plane have not a com- 
mon intersection with those of both orbits, it is equally easy 
to see that the node of the disturbed orbit may either recede 
on both that plane and the disturbing orbit or advance on 
the one and recede on the other, according to the relative 
situation of the planes. 

(629.) This is the case with the planetary orbits. They 
do not all intersect each other in a common node. Although 
perfectly true, therefore, that the node of any one planet 
would recede on the orbit of any and each other by the 
individual action of that other, yet, when all act together, 
recess on one plane may be equivalent to advance on 
another, so that the motion of the node of any one orbit 
on a given plane, arising from their joint action, taking 
into account the different situations of all the planes, be- 
comes a curiously complicated phenomenon whose law 
cannot be very easily expressed in words, though reducible 
to strict numerical statement, being, in fact, a mere geomet- 
rical result of what is above shown. 

(630.) The nodes of all the planetary orbits on the true 
ecliptic, as a matter of fact, are retrograde, though they are 
not all so on a fixed plane, such as we may conceive to 
exist in the planetary system, and to be a plane of reference 
unaffected by their mutual disturbances. It is, however, 
to the ecliptic, that we are under the necessity of referring 



588 OUTLINES OF ASTRONOMY 

their movements from oar station in the system; and if we 
would transfer our ideas to a fixed plane, it becomes neces- 
sary to take account of the variation of the ecliptic itself, 
produced by the joint action of all the planets. 

(631.) Owing to the smallness of the masses of the planets 
and their great distances from each other, the revolutions of 
their nodes are excessively slow, being in every case less 
than a single degree per century, and in most cases not 
amounting to half that quantity. It is otherwise with the 
moon, and that owing to two distinct reasons. First, that 
the disturbing force itself arising from the sun's action (as 
appears from the table given in art. 612) bears a much larger 
proportion to the earth's central attraction on the moon than 
in the case of any planet disturbed by any other. And 
secondly, because the synodic revolution of the moon, 
within which the average is struck (and always on the 
side of recess), is only 29 L days, a period much shorter than 
that of any of the planets, and vastly so than that of several 
among them. All this, is agreeable to what has already been 
stated (arts. 407, 408) respecting the motion of the moon's 
nodes, and it is hardly necessary to mention that, when cal- 
culated, as it has been, d priori, from an exact estimation 
of all the acting forces, the result is found to coincide with 
perfect precision with that immediately derived from obser- 
vation, so that not a doubt can subsist as to this being the 
real process by which so remarkable an effect is produced. 

(632.) So far as the physical condition of each planet is 
concerned, it is evident that the position of their nodes can 
be of little importance. It is otherwise with the mutual 
inclinations of their orbits with .respect to each other, and 
to the equator of each. A variation in the position of the 
ecliptic, for instance, by which its pole should shift its dis- 



OUTLINES OF ASTRONOMY 589 

tance from the pole of the equator, would disturb our 
seasons. Should the plane of the earth's orbit, for instance, 
ever be so changed as to bring the ecliptic to coincide with 
the equator, we should have perpetual spring over all the 
world; and, on the other hand, should it coincide with 
a meridian, the extremes of summer and winter would be- 
come intolerable. The inquiry, then, of the variations of 
inclination of the planetary orbits inter se, is one of much 
higher practical interest than those of their nodes. 

(633.) Referring to the figures of art. 610 et seq., it is 
evident that the plane S P q, in which the disturbed body 
moves during an instant of time from its quitting P, is 
differently inclined to the orbit of M, or to a fixed plane, 
from the original or undisturbed plane P S p. The differ- 
ence of absolute position of these two planes in space is the 
angle made between the planes P S R and P S r, and is 
therefore calculable by spherical trigonometry, when the 
angle R S r or the momentary recess of the node is known, 
and also the inclination of the planes of the orbits to each 
other. We perceive, then, that between the momentary 
change of inclination and the momentary recess of the 
node there exists an intimate relation, and that the research 
of the one is in fact bound up in that of the other. This 
may be, perhaps, made clearer, by considering the orbit of 
P to be not merely an imaginary line, but an actual circle 
or elliptic hoop of some rigid material, without inertia, on 
which, as on a wire, the body P may slide as a bead. It 
is evident that the position of this hoop will be determined 
at any instant, by its inclination to the ground plane to 
which it is referred, and by the place of its intersection 
therewith, or node. It will also be determined by the 
momentary direction cf P's motion, which (having no 



540 OUTLINES OF ASTRONOMY 

inertia) it must obey ; and any change by which P should, 
in the next instant, alter its orbit, would be equivalent to 
a shifting, bodily, of the whole hoop, changing at once its 
inclination and nodes. 

(634.) One immediate conclusion from what has been 
pointed out above, is that where the orbits, as in the case 
of the planetary system and the moon, are slightly inclined 
to one another, the momentary variations of the inclination 
are of an order much inferior in magnitude to those in the 
place of the node. This is evident on a mere inspection of 
our figure, the angle R P r being, by reason of the small 
inclination of the planes S P R and R S r, necessarily 
much smaller than the angle RSr. In proportion as the 
planes of the orbits are brought to coincidence, a very 
trifling angular movement of V p about P S as an axis will 
make a great variation in the situation of the point r, where 
its prolongation intersects the ground plane. 

(635.) Referring to the figure of art. 622, we perceive 
that although the motion of the node is retrograde whenever 
the momentary disturbed arc P Q lies between the planes 
C A and C G A of the two orbits, and vice versa, indiffer- 
ently whether P be in the act of receding from the plane 
C A, as in the quadrant C Gr, or of approaching to it, as 
in G A, yet the same identity as to the character of tho 
change does not subsist in respect of the inclination. The 
inclination of the disturbed orbit (i.e. of its momentary 
element) P q or P q\ is measured by the spherical angle 
PrHorPr'H. Now in the quadrant C Gr, P r H is less, 
and Pr'H greater than PCS; but in G A, the converse. 
Hence this rule: 1st, If the disturbing force urge P toward 
the plane of M's orbit, and the undisturbed motion of P carry 
it also toward that plane; and 2dly, if the disturbing force 



OUTLINES OF ASTRONOMY 541 

urge P from that plane, while P's undisturbed motion also 
carries it from it, in either case the inclination momentarily 
increases; but if, 3dly, the disturbing force act to, and P's 
motion carry it from — or if the force act from, and the mo- 
tion carry it to, that plane, the inclination momentarily 
diminishes. Or (including all the cases under one alter- 
native) if the action of the disturbing force and the undis- 
turbed motion of P with reference to the plane of M's orbit 
be of the same character, the inclination increases; if of 
contrary characters, it diminishes. 

(636.) To pass from the momentary changes which take 
place in the relations of nature to the accumulated effects 
produced in considerable lapses of time by the continued 
action of the same causes, under circumstances varied by 
these very effects, is the business of the integral calculus. 
Without going into any calculations, however, it will be 
easy for us to demonstrate, from the principles above laid 
down, the leading features of this part of the planetary 
theory, viz. the periodic nature of the change of the in- 
clinations of two orbits to each other, the re-establishment 
of their original values, and the consequent oscillation of 
each plane about a certain mean position. As in explain- 
ing the motion of the nodes, we will commence, as the 
simplest case, with that of an exterior planet disturbed by 
an interior one at less than half its distance from the central 
body. Let A C A' be the great circle of the heavens into 
which M's orbit seen from S is projected, extended into a 
straight line, and A g C h A' the corresponding projection 
of the orbit of P so seen. Let M occupy some fixed situa- 
tion, suppose in the semicircle A C, and let P describe a 
complete revolution from A through g C h to A'. Then 
while it is between A and g or in its first quadrant, its mo- 



542 OUTLINES OF ASTRONOMY 

tion is from the plane of M's orbit, and at the same time 
the orthogonal force acts from that plane: the inclination, 
therefore (art. 635) increases. In the second quadrant the 
motion of P is to, but the force continues to act from, 
the plane, and the inclination again decreases. A similar 
alternation takes place in its course through the quadrants 
C h and h A. Thus the plane of P's orbit oscillates to and 
fro about its mean position twice in each revolution of P. 
During this process if M held a fixed position at G, the 
forces being symmetrically alike on either side, the extent 
of these oscillations would be exactly equal, and the inclina- 
tion at the end of one revolution of P would revert precisely 
to its original value. But if M be elsewhere, this will not 
be the case, and in a single revolution of P, only a partial 



compensation will be operated, and an overplus on the side, 
suppose of diminution, will remain outstanding. But when 
M comes to M', a point equidistant from Gr on the other 
side, this effect will be precisely reversed (supposing the 
orbits circular). On the average of both situations, there- 
fore, the effect will be the same as if M were divided into 
two equal portions, one placed at M and the other at M', 
which will annihilate the preponderance in question and 
effect a perfect restoration. And on an average of all pos- 
sible situations of M, the effect will in like manner be 
the same as if its mass were distributed over the whole 
circumference of its orbit, forming a ring, each portion of 
which will exactly destroy the effect of that similarly 
situated on the opposite side of the line of nodes. 



OUTLINES OF ASTRONOMY 513 

(687.) The reasoning is precisely similar for the more 
complicated cases of arts. 625 and 627. Suppose that 
owing either to the proximity of the two orbits (in the case 
of an exterior disturbed planet) or to the disturbed orbit 
being interior to the disturbing one, there were a larger or 
less portion, d e, of P's orbit in which these relations were 
reversed. Let M be the position of M' corresponding to 
d e, then taking Gr W = Gr M, there will be a similar portion 
d' e' bearing precisely the same reversed relation to M', and 
therefore, the actions of M' M will equally neutralize each 
other in this as in the former state of things. 

(638.) To operate a complete and rigorous compensation, 
however, it is necessary that M should be presented to P in 
every possible configuration, not only with respect to P 
itself, but to the line of nodes, to the position of which 
line the whole reasoning bears reference. In the case of 
the moon, for example, the disturbed body (the moon) 
revolves in 27 d> 322, the disturbing (the sun) in 365 d -256, and 
the line of nodes in 6793 d *391, numbers in proportion to 
each other about as 1 to 13 and 249 respectively. Now in 
13 revolutions of P, and one of M, if the node remained 
fixed, P would have been presented to M so nearly in every 
configuration as to operate an almost exact compensation. 
But in 1 revolution of M, or 13 of P, the node itself has 
shifted ,S or about J of a revolution, in a direction opposite 
to the revolutions of M and P, so that although P has been 
brought back to the same configuration with respect to M, 
both are J of a revolution in advance of the same configura- 
tion as respects the node. The compensation, therefore, 
will not be exact, and to make it so, this process must be 
gone through 19 times, at the end of which both the bodies 
will be restored to the same relative position, not only with 



544 OUTLINES OF ASTRONOMY 

respect to each other, but to the node. The fractional parts 
of entire revolutions, which in this explanation have been 
neglected, are evidently no further influential than as ren- 
dering the compensation thus operated in a revolution of 
the node slightly inexact, and thus giving rise to a com- 
pound period of greater duration, at the end of which a 
compensation almost mathematically rigorous will have 
been effected. 

(639.) It is clear then, that if the orbits be circles, the 
lapse of a very moderate number of revolutions of the bodies 
will very nearly, and that of a revolution of the node almost 
exactly, bring about a perfect restoration of the inclinations. 
If, however, we suppose the orbits excentric, it is no less 
evident, owing to the want of symmetry in the distribution 
of the forces, that a perfect compensation will not be effected 
either in one or in any number of revolutions of P and M, 
independent of the motion of the node itself, as there will 
always be some configuration more favorable to either an in- 
crease of inclination than its opposite is unfavorable. Thus 
will arise a change of inclination which, were the nodes and 
apsides of the orbits fixed, would be always progressive in 
one direction until the planes were brought to coincidence. 
But, 1st, half a revolution of the nodes would of itself re- 
verse the direction of this progression by making the posi- 
tion in question favor the opposite movement of inclination; 
and, 2dly, the planetary apsides are themselves in motion 
with unequal velocities, and thus the configuration whose 
influence destroys the balance, is, itself, always shifting its 
place on the orbits. The variations of inclination dependent 
on the excentricities are therefore, like those independent of 
them, periodical, and being, moreover, of an order more 
minute (by reason of the smallness of the excentricities) 



OUTLINES OF ASTRONOMY 545 

than the latter, it is evident that the total variation of the 
planetary inclinations must fluctuate within very narrow 
limits. Geometers have accordingly demonstrated by an 
accurate analysis of all the circumstances, and an exact esti- 
mation of the acting forces, that such is the case; and this 
is what is meant by asserting the stability of the planetary 
system as to the mutual inclinations of its orbits. By the 
researches of Lagrange (of whose analytical conduct it is 
impossible here to give any idea), the following elegant 
theorem has been demonstrated: — 

"If the mass of every planet be multiplied by the square root 
of the major axis of its orbit, and the product by the square of 
the tangent of its inclination to a fixed plane, the sum of all 
these products will be constantly the same under the influence 
of their mutual attraction." If the present situation of the 
plane of the ecliptic be taken for that fixed plane (the eclip- 
tic itself being variable like the other orbits), it is found that 
this sum is actually very small: it must, therefore, always 
remain so. This remarkable theorem alone, then, would 
guarantee the stability of the orbits of the greater planets; 
but from what has above been shown of the tendency of each 
planet to work out a compensation on every other, it is evi- 
dent that the minor ones are not excluded from this benefi- 
cial arrangement. 

(640.) Meanwhile, there is no doubt that the plane of the 
ecliptic does actually vary hj the actions of the planets. 
The amount of this variation is about 48" per century, and 
has long been recognized by astronomers, by an increase of 
the latitudes of all the stars in certain situations, and their 
diminution in the opposite regions. Its effect is to bring the 
ecliptic by so much per annum nearer to coincidence with 
the equator; but from what we have above seen, this dimi- 



546 OUTLINES OF ASTRONOMY 

nution of the obliquity of the ecliptic will not go on beyond 
certain very moderate limits, after which (although in an im- 
mense period of ages, being a compound cycle resulting from 
the joint action of all the planets) it will again increase, and 
thus oscillate backward and forward about a mean position, 
the extent of its deviation to one side and the other being 



than 1° 21'. 

(641.) One effect of this variation of the plane of the 
ecliptic — that which causes its nodes on a fixed plane to 
change — is mixed up with the precession of the equinoxes, 
and indistinguishable from it, except in theory. This last- 
mentioned phenomenon is, however, due to another cause, 
analogous, it is true, in a general point of view, to those 
above considered, but singularly modified by the circum- 
stances under which it is produced. We shall endeavor to 
render these modifications intelligible, as far as they can be 
made so without the intervention of analytical formulae. 

(642.) The precession of the equinoxes, as we have shown 
in art. 312, consists in a continual retrogradation of the node 
of the earth's equator on the ecliptic; and is, therefore, ob- 
viously an effect so far analogous to the general phenomenon 
of the retrogradation of the nodes of the orbits on each other. 
The immense distance of the planets, however, compared 
with the size of the earth, and the smallness of their masses 
compared to that of the sun, puts their action out of the 
question in the inquiry of its cause, and we must, therefore, 
look to the massive though distant sun, and to our near 
though minute neighbor, the moon, for its explanation. 
This will, accordingly, be found in their disturbing action 
on the redundant matter accumulated on the equator of the 
earth, by which its figure is rendered spheroidal, combined 
with the earth's rotation on its axis. It is to the sagacity of 



OUTLINES OF ASTRONOMY 547 

Newton that we owe the discovery of this singular mode of 
action. 

(643.) Suppose in our figure (art. 611) that instead of one 
body, P, revolving round S, there were a succession of par- 
ticles not coherent, but forming a kind of fluid ring, free to 
change its form by any force applied. Then, while this ring 
revolved round S in its own plane, under the disturbing in- 
fluence of the distant body M (which now represents the 
moon or the sun, as P does one of the particles of the earth's 
equator), two things would happen: 1st, its figure would be 
bent out of a plane into an undulated form, those parts of it 
within the arcs D A and E C being rendered more inclined 
to the plane of M's orbit, and those within the arcs A E, 
C D, less so than they would otherwise be; 2dly, the nodes 
of this ring, regarded as a whole, without respect to its 
change of figure, would retreat upon that plane. 

(644.) But suppose this ring, instead of consisting of dis- 
crete molecules free to move independently, to be rigid and 
incapable of such flexure, like the hoop we have supposed in 
art. 633, but having inertia, then it is evident that the effort 
of those parts of it which tend to become more inclined will 
act through the medium of the ring itself (as a mechanical 
engine or lever) to counteract the effort of those which have 
at the same instant a contrary tendency. In so far only, 
then, as there exists an excess on the one or the other side 
will the inclination change, an average being struck at every 
moment of the ring's motion; just as was shown to happen 
in the view we have taken of the inclinations, in every com- 
plete revolution of a single disturbed body, under the influ- 
ence of a fixed disturbing one. 

(645.) Meanwhile, however, the nodes of the rigid ring 
will retrograde, the general or average tendency of the nodes 



548 OUTLINES OF ASTRONOMY 

of every molecule being to do so. Here, as in the other 
case, a struggle will take place by the counteracting efforts 
of the molecules contrarily disposed, propagated through the 
solid substance of the ring; and thus at every instant of 
time, an average will be struck, which being identical in its 
nature with that effected in the complete revolution of a sin- 
gle disturbed body, will, in every case, be in favor of a re- 
cess of the node, save only when the disturbing body, be it 
sun or moon, is situated in the plane of the earth's equator. 

(646.) This reasoning is evidently independent of any 
consideration of the cause which maintains the rotation of 
the ring; whether the particles be small satellites retained in 
circular orbits under the equilibrated action of attractive and 
centrifugal forces, or whether they be small masses conceived 
as attached to a set of imaginary spokes, as of a wheel, cen- 
tring in S, and free only to shift their planes by a motion of 
those spokes perpendicular to the plane of the wheel. This 
makes no difference in the general effect; though the differ- 
ent velocities of rotation, which may be impressed on such a 
system, may and will have a very great influence both on 
the absolute and relative magnitudes of the two effects in 
question— the motion of the nodes and change of inclination. 
This will be easily understood, if we suppose the ring with- 
out a rotatory motion, in which extreme case it is obvious 
that so long as M remained fixed there would take place no 
recess of nodes at all, but only a tendency of the ring to tilt 
its plane round a diameter perpendicular to the position of 
M, bringing it toward the line S M. 

(647.) The motion of such a ring, then, as we have been 
considering, would imitate, so far as the recess of the nodes 
goes, the precession of the equinoxes, only that its nodes 
would retrograde far more rapidly than the observed preces- 



OUTU. ■ ASTRONOMY 549 

sion, which is excessively slow. But now conceive this ring 
to be loaded with a spherical mass enormously heavier than 
itself, placed concentrically within it, and cohering firmly, to 
it, but indifferent, or very nearly so, to any such cause of 
motion; and suppose, moreover, that instead of one such 
ring there are a vast multitude heaped together around the 
equator of such a globe, so as to form an elliptical protuber- 
ance, enveloping it like a shell on all sides, but whose mass, 
taken together, should form but a very minute fraction of 
the whole spheroid. We have now before us a tolerable rep- 
resentation of the case of nature; 5 and it is evident that the 
rings, having to drag round with them in their nodal revolu- 
tion this great inert mass, will have their velocity of retro- 
gradation proportionally diminished. Thus, then, it is easy 
to conceive how a motion similar to the precession of the 
equinoxes, and, like it, characterized by extreme slowness, 
will arise from the causes in action. It may seem at first 
sight paradoxical that the whole effect of the external attrac- 
tion should terminate in the production of such a movement, 
without producing any change in the inclination of the 
equator to the ecliptic. But a due consideration of the rea- 
soning in arts. 636, 637 will make it evident that for every 
particle in the revolving ring (in every situation of the dis- 
turbing body) whose change of motion would tend to create 

5 That a perfect sphere would be so inert and indifferent as to a revolution 
of the nodes of its equator under the influence of a distant attracting body ap- 
pears from this — that the direction of the resultant attraction of such a body, 
or of that single force which, opposed, would neutralize and destroy its whole 
action, is necessarily in a line passing through the centre of the sphere, and, 
therefore, can have no tendency to turn the sphere one way or other. It may 
be objected by the reader, that the whole sphere may be conceived as consist- 
ing of rings parallel to its equator, of every possible diameter, and that, there- 
fore, it3 nodes should retrograde even without a protuberant equator. The in- 
ference is incorrect, but our limits will not allow us to go into an exposition of 
the fallacy. We should, however, caution him, generally, that no dynamical 
subject is open to more mistakes of this kind, which nothing but the closest 
attention, in every varied point of view, will detect. 



550 OUTLINES OF ASTRONOMY 

a change of inclination in one direction, there exists another, 
exercising an equal tendency of an opposite kind. 

(648.) Now a recess of the node of the earth's equator, 
upon a given plane, corresponds to a conical motion of its 
axis round a perpendicular to that plane. But in the case 
before us, that plane is not the ecliptic, but the moon's orbit 
for the time being; and it may be asked how we are to rec- 
oncile this with what is stated in art. 317 respecting the na- 
ture of the motion in question. To this we reply, that the 
nodes of the lunar orbit, being in a state of continual and 
rapid retrogradation, while its inclination is preserved nearly 
invariable, the point in the sphere of the heavens round 
which the pole of the earth's equator revolves (with that 
extreme slowness characteristic of the precession) is itself in 
a state of continual circulation round the pole of the eclip- 
c__^^ tic, with that much more rapid motion 

/ \ \ which belongs to the lunar node. A 

j B glance at the annexed figure will ex- 

\g// plain this better than words. P is the 

\\ / pole of the ecliptic, A the pole of the 

\ I moon's orbit, moving round the small 

\\/ \ circle A B C D in 19 years; a the pole 

/k V^^' of the earth's equator, which at each mo- 
e bed ment of its progress has a direction perpen- 

dicular to the varying position of the line A a, and a velocity 
depending on the varying intensity of the acting causes dur- 
ing the period of the nodes. This velocity, however, being 
extremely small, when A comes to B, C, D, E, the line A a 
will have taken up the positions B b, C c, D d, E e, and the 
earth's pole a will thus, in one tropical revolution of the 
node, have arrived at e, having described not an exactly cir- 
cular arc a e, but a single undulation of a wave-sbape or 



OUTLINES OF ASTRONOMY 551 

epicycloidal curve, abode, with a velocity alternately 
greater and less than its mean motion, and this will be re- 
peated in every succeeding revolution of the node. 

(619.) Now this is precisely the kind of motion which, as 
we have seen in art. 325, the pole of the earth's equator 
really has round the pole of the ecliptic, in consequence of 
the joint effects of precession and nutation, which are thus 
uranographically represented. If we superadd to the effect 
of lunar precession that of the solar, which alone would 
cause the pole to describe a circle uniformly about P, this 
will only affect the undulations of our waved curve, by ex- 
tending them in length, but will produce no effect on the 
depth of the waves, or the excursions of the earth's axis to 
and from the pole of the ecliptic. Thus we see that the two 
phenomena of nutation and precession are intimately con- 
nected, or rather both of them essential constituent parts of 
one and the same phenomenon. It is hardly necessary to 
state that a rigorous analysis of this great problem, by an 
exact estimation of all the acting forces and summation of 
their dynamical effects, leads to the precise value of the 
coefficients of precession and nutation, which observation 
assigns to them. The solar and lunar portions of the pre- 
cession of the equinoxes, that is to say, those portions which 
are uniform, are to each other in the proportion of about 
2 to 5. 

(650.) In the nutation of the earth's axis we have an ex- 
ample (the first of its kind which has occurred to us) of a 
periodical movement in one part of the system, giving rise 
to a motion having the same precise period in another. The 
motion of the moon's nodes is here, we see, represented, 
though under a very different form, yet in the same exact 
periodic time, by a movement of a peculiar oscillatory kind 



552 OUTLINES OF ASTRONOMY 

impressed on the solid mass of the earth. We must not let 
the opportunity pass of generalizing the principle involved 
in this result, as it is one which we shall find again and again 
exemplified in every part of physical astronomy, nay, in 
every department of natural science. It may be stated as 
"the principle of forced oscillations, or of forced vibra- 
tions," and thus generally announced: — 

If one part of any system connected either by material ties, 
or by the mutual attractions of its members, be continually 
maintained by any cause, whether inherent in the constitution 
of the system or external to it, in a state of regular periodic 
motion, that motion will be propagated throughout the whole 
system and will give rise, in every member of it, and in every 
part of each member, to periodic movements executed in equal 
period, with that to which they owe their origin, though not nec- 
essarily synchronous with them in their maxima and minima* 

The system may be favorably or unfavorably constituted 
for such a transfer of periodic movements, or favorably in 
some of its parts and unfavorably in others; and accordingly 
as it is the one or the other, the derivative oscillation (as it 
may be termed) will be imperceptible in one case, of appre- 
ciable magnitude in another, and even more perceptible in 
its visible effects than the original cause in a third; of this 
last kind we have an instance in the moon's acceleration, to 
be hereafter noticed. 

(651.) It so happens that our situation on the earth, and 
the delicacy which our observations have attained, enable us 
to make it as it were an instrument to feel these forced vibra- 
tions — these derivative motions, communicated from various 

6 See a demonstration of this theorem for the forced vibrations of system, 
connected by material ties of imperfect elasticity, in my treatise on Sound, 
Encyc. Metrop. art. 323. The demonstration is easily extended and general- 
ized to take in other systems. 



OUTLINES OF ASTRONOMY 553 

quarters, especially from our near neighbor, the moon, much 
in the same way as we detect, by the trembling of a board 
beneath us, the secret transfer of motion by which the 
sound of an organ pipe is dispersed through the air, and car- 
ried down into the earth. Accordingly, the monthly revo- 
lution of the moon, and the annual motion of the sun, pro- 
duce, each of them, small nutations in the earth's axis, 
whose periods are respectively half a month and half a year, 
each of which, in this view of the subject, is to be regarded 
as one portion of a period consisting of two equal and simi- 
lar parts. But the most remarkable instance, by far, of this 
propagation of periods, and one of high importance to man- 
kind, is that of the tides, which are forced oscillations, ex- 
cited by the rotation of the earth in an ocean disturbed from 
its figure by the varying attractions of the sun and moon, 
each revolving in its own orbit, and propagating its own 
period into the joint phenomenon. The explanation of the 
tides, however, belongs more properly to that part of the 
general subject of perturbations which treats of the action 
of the radial component of the disturbing force, and is there- 
fore postponed to a subsequent chapter. 



554 OUTLINES OF ASTRONOMY 



CHAPTER XIII 

THEORY OF THE AXES, PERIHELIA, AND ECCENTRICITIES 
"Incipiunfc magni procedere menses." — Virg. Pollio. 

Variation of Elements in G-enerai — Distinction Between Periodic and Sec- 
ular Variations — Geometrical Expression of Tangential and Normal 
Forces — Variation of the Major Axis Produced only by the Tangential 
Force— Lagrange's Theorem of the Conservation of the Mean Dis- 
tances and Periods — Theory of the Perihelia and Excentricities — 
Geometrical Representation of their Momentary Variations — Estima- 
tion of the Disturbing Forces in Nearly Circular Orbits — Application 
to the Case of the Moon — Theory of the Lunar Apsides and Excen- 
tricity — Experimental Illustration — Application of the Foregoing Prin- 
ciples to the Planetary Theory — Compensation in Orbits very Nearly 
Circular — Effects of Ellipticity — General Results — Lagrange's Theorem 
of the Stability of the Excentricities 

(652.) In the foregoing chapter we have sufficiently 
explained the action of the orthogonal component of the 
disturbing force, and traced it to its results in a continual 
displacement of the plane of the disturbed orbit, in virtue 
of which the nodes of that plane alternately advance and 
recede upon the plane of the disturbing body's orbit, with 
a general preponderance on the side of advance, so as after 
the lapse of a long period to cause the nodes to make a com- 
plete revolution and come around to their former situation. 
At the same time the inclination of the plane of the dis- 
turbed motion continually changes, alternately increasing 
and diminishing; the increase and diminution however com- 
pensating each other, nearly in single revolutions of the 
disturbed and disturbing bodies, more exactly in many, 
and with perfect accuracy in long periods, such as those 



OUTLINES OP' ASTRONOMY 555 

of a complete revolution of the nodes and apsides. In the 
present and following chapters we shall endeavor to trace 
the effects of the other components of the disturbing force — 
those which act in the plane (for the time being) of the dis- 
turbed orbit, and which tend to derange the elliptic form of 
the orbit, and the laws of elliptic motion in that plane. 
The small inclination, generally speaking, of the orbits 
of the planets and satellites to each other, permits us to 
separate these effects in theory one from the other, and 
thereby greatly to simplify their consideration. Accord- 
ingly, in what follows, we shall throughout neglect the 
mutual inclination of the orbits of the disturbed and dis- 
turbing bodies, and regard all the forces as acting and all 
the motions as performed in one plane. 

(653.) In considering the changes induced by the mutual 
action of two bodies in different aspects with respect to each 
other on the magnitudes and forms of their orbits and in 
their positions therein, it will be proper in the first instance 
to explain the conventions under which geometers and 
astronomers have alike agreed to use the language and laws 
of the elliptic system, and to continue to apply them to 
disturbed orbits, although those orbits so disturbed are no 
longer, in mathematical strictness, ellipses, or any known 
curves. This they do, partly on account of the convenience 
of conception and calculation which attaches to this system, 
but much more for this reason — that it is found, and may 
be demonstrated from the dynamical relations of the case, 
that the departure of each planet from its ellipse, as deter- 
mined at any epoch, is capable of being truly represented, 
by supposing the ellipse itself to be slowly variable, to 
change its magnitude and excentricity, and to shift its 

position and the plane in -which it lies according to certain 

Astronomy — Vol. XX — 5 



556 OUTLINES OF ASTRONOMY 

laws, while the planet all the time continues to move in 
this ellipse, just as it would do if the ellipse remained 
invariable and the disturbing forces had no existence. By 
this way of considering the subject, the whole effect of the 
disturbing forces is regarded as thrown upon the orbit, 
while the relations of the planet to that orbit remain un- 
changed. This course of procedure, indeed, is the most 
natural, and is in some sort forced upon us by the extreme 
slowness with which the variations of the elements, at least 
where the planets only are concerned, develop themselves. 
For instance, the fraction expressing the excentricity of the 
earth's orbit changes no more than 0-00004 in its amount in 
a century; and the place of its perihelion, as referred to the 
sphere of the heavens, by only 19' 39" in the same time. 
For several years, therefore, it would be next to impossible 
to distinguish between an ellipse so varied and one that had 
not varied at all; and in a single revolution, the difference 
between the original ellipse and the curve really represented 
by the varying one, is so excessively minute, that, if ac- 
curately drawn on a table, six feet in diameter, the nicest 
examination with microscopes, continued along the whole 
outlines of the two curves, would hardly detect any per- 
ceptible interval between them. Not to call a motion so 
minutely conforming itself to an elliptic curve, elliptic, 
would be affectation, even granting the existence of trivial 
departures alternately on one side or on the other; though, 
on the other hand, to neglect a variation, which continues 
to accumulate from age to age, till it forces itself on our 
notice, would be wilful blindness. 

(654.) Geometers, then, have agreed in each single revo- 
lution, or for any moderate interval of time, to regard the 
motion of each planet as elliptic, and performed according 



OUTLINES OF ASTRONOMY 557 

to Kepler's iaws, with a reserve in favor of those very small 
and transient fluctuations which take place within that time, 
but at the same time to regard ail the elements of each ellipse 
as in a continual, though extremely slow, state of change; 
and, in tracing the effects of perturbation on the system, 
they take account principally, or entirely, of this change of 
the elements, as that upon which any material change in the 
great features of the system will ultimately depend. 

(6od.) And here we encounter the distinction between 
what are termed secular variations, and such as are rapidly 
periodic, and are compensated in short intervals. In our 
exposition of the variation of the inclination of a disturbed 
orbit (art. 636), for instance, we showed that, in each single 
revolution of the disturbed body, the plane of its motion 
underwent fluctuations to and fro in its inclination to that 
of the disturbing body, which nearly compensated each 
other; leaving, however, a portion outstanding, which again 
is nearly compensated by the revolution of the disturbing 
body, yet still leaving outstanding and uncompensated a 
minute portion of the change which requires a whole revo- 
lution of the node to compensate and bring it back to an 
average or mean value. Now, the first two compensations 
which are operated by the planets going through the suc- 
cession of configurations with each other, and therefore in 
comparatively short periods, are called periodic variations; 
and the deviations thus compensated are called inequalities 
depending on. configurations ; while the last, which is oper- 
ated by a period of the node (one of the elements), has noth- 
ing to do with the configurations of the individual planets, 
requires a very long period of time for its consummation, 
and is, therefore, distinguished from the former by the term 
secular variation. 



558 OUTLINES OF ASTRONOMY 

(656.) It is true, that, to afford an exact representation 
of the motions of a disturbed body, whether planet or 
satellite, both periodical and secular variations, with their 
corresponding inequalities, require to be expressed; and, 
indeed, the former even more than the latter; seeing that 
the secular inequalities are, in fact, nothing but what re- 
mains after the mutual destruction of a much larger amount 
(as it very often is) of periodical. But these are in their 
nature transient and temporary: they disappear in short 
periods, and leave no trace. The planet is temporarily 
drawn from its orbit (its slowly varying orbit), but forth- 
with returns to it, to deviate presently as much the other 
way, while the varied orbit accommodates and adjusts itself 
to the average of these excursions on either side of it; and 
thus continues to present, for a succession of indefinite ages, 
a kind of medium picture of all that the planet has been 
doing in their lapse, in which the expression and character 
is preserved; but the individual features are merged and 
lost. These periodic inequalities, however, are, as we have 
observed, by no means neglected, but it is more convenient 
to take account of them by a separate process, independent 
of the secular variations of the elements. 

(657.) In order to avoid complication, while endeavoring 
to give the reader an insight into both kinds of variations, 
we shall for the present conceive all the orbits to lie in one 
plane, and confine our attention to the case of two only, that 
of the disturbed and disturbing body, a view of the subject 
which (as we have seen) comprehends the case of the moon 
disturbed by the sun, since any one of the bodies may be 
regarded as fixed at pleasure, provided we conceive all its 
motions transferred in a contrary direction to each of the 
others. Let therefore A P B be the undisturbed elliptic 



OUTLINES OF ASTRONOMY 



559 



orbit of a planet P; M a disturbing body, join M P, and 
supposing M K = M S take M N : M K :: M K" : M P 3 . 
Then if SN be joined, N S will represent the disturbing 
force of M on P, on the same scale that S M represents M's 
attraction on S. Suppose Z P Y a tangent at P, S Y per- 
pendicular to it, and NT,N L perpendicular respectively 
to S Y and P S produced. Then will N T represent the 
tangential, T S the normal, N" L the transversal, and L S 
the radial components of the disturbing force. In circular 
orbits or orbits only slightly elliptic, the directions P S L 




and S Y are nearly coincident, and the former pair of forces 
will differ but slightly from the latter. We shall here, how- 
ever, take the general case, and proceed to investigate in an 
elliptic orbit of any degree of excentricity the momentary 
changes produced by the action of the disturbing force in 
those elements on which the magnitude, situation, and form, 
of the orbit depend {i.e. the length and position of the major 
axis and the excentricity), in the same way as in the last 
chapter we determined the momentary changes of the in- 
clination and node similarly produced by the orthogonal 
force. 

(658.) We shall begin with the momentary variation in 



560 OUTLINES OF ASTRONOMY 

the length of the axis, an element of the first importance, 
as on it depend (art. 487) the periodic time and mean angu- 
lar motion of the planet, as well as the average supply of 
light and heat it receives in a given time from the sun, any- 
permanent or constantly progressive change in which would 
alter most materially the conditions of existence of living 
beings on its surface. Now it is a property of elliptic 
motion performed under the influence of gravity, and in 
conformity with Kepler's laws, that if the velocity with 
which a planet moves at any point of its orbit be given, 
and also the distance of that point from the sun, the major 
axis of the orbit is thereby also given. It is no matter in 
what direction the planet may be moving at that moment. 
This will influence the excentricity and the position of its 
ellipse, but not its length. This property of elliptic motion 
has been demonstrated by Newton, and is one of the most 
obvious and elementary conclusions from his theory. Let 
us now consider a planet describing an indefinitely small 
arc of its orbit about the sun, under the joint influence of 
its attraction, and the disturbing power of another planet. 
This arc will have some certain curvature and direction, 
and, therefore, may be considered as an arc of a certain 
ellipse described about the sun as a focus, for this plain 
reason — that whatever be the curvature and direction of 
the arc in question, an ellipse may always be assigned, 
whose focus shall be in the sun, and which shall coincide 
with it throughout the whole interval (supposed indefi- 
nitely small) between its extreme points. This is a matter 
of pure geometry. It does not follow, however, that the 
ellipse thus instantaneously determined will have the same 
elements as that similarly determined from the arc described 
in either the previous or the subsequent instant. If the dis- 



OUTLINES OF ASTRONOMY 561 

turbing force did not exist, this would be the case; but, by 
its action, a variation of the element from instant to instant 
is produced, and the ellipse so determined is in a continual 
state of change. Now when the planet has reached the end 
of the small arc under consideration, the question whether 
it will in the next instant describe an arc of an ellipse hav- 
ing the same or a varied axis will depend, not on the new 
direction impressed upon it by the acting forces — for the 
axis, as we have seen, is independent of that direction — 
not on its change of distance from the sun, while describing 
the former arc — for the elements of that arc are accommo- 
dated to it, so that one and the same axis must belong to 
its beginning and its end. The question, in short, whether 
in the next arc it shall take up a new major axis or go 
on with the old one will depend solely on this — whether 
its velocity has or has not undergone a change by the ac- 
tion of the disturbing force. For the central force residing 
in the focus can impress on it no such change of velocity 
as to be incompatible with the permanence of its ellipse, 
seeing that it is by the action of that force that the velocity 
is maintained in that due proportion to the distance which 
elliptic motion, as such, requires. 

(659.) Thus we see that the momentary variation of the 
major axis depends on nothing but the momentary deviation 
from the law of elliptic velocity produced by the disturbing 
force, without the least regard to the direction in which that 
extraneous velocity is impressed, or the distance from the 
sun at which the planet may be situated, at the moment of 
its impression. Nay, we may even go further, for, as this 
holds good at every instant of its motion, it will follow, 
that after the lapse of any time, however great, the total 
amount of change which the axis may have undergone will 



562 OUTLINES OF ASTRONOMY 

be determined only by the total deviation produced by the 
action of the disturbing force in the velocity of the dis- 
turbed body from that which it would have had in its un- 
disturbed ellipse, at the same distance from the centre, and 
that therefore the total amount of change produced in the 
axis in any lapse of time may be estimated, if we know at 
every instant the efficacy of the disturbing force to alter 
the velocity of the body's motion, and that without any 
regard to the alterations which the action of that force 
may have produced in the other elements of the motion 
in the same time. 

(660.) Now it is not the whole disturbing force which is 
effective in changing P's velocity, but only its tangential 
component. The normal component tends merely to alter 
the curvature of the orbit or to deflect it into conformity 
with a circle of curvature of greater or lesser radius, as the 
case may be, and in no way to alter the velocity. Hence 
it appears that the variation of the length of the axis is due 
entirely to the tangential force, and is quite independent on the 
normal. Now it is easily shown that as the velocity in- 
creases, the axis increases (the distance remaining unal- 
tered 1 ) though not in the same exact proportion. Hence 
it follows that if the tangential disturbing force conspires 
with the motion of P, its momentary action increases the 
axis of the disturbed orbit, whatever be the situation 
of P in its orbit, and vice versa. 

(661.) Let A S B {fig. art. 657) be the major axis of the 
ellipse A P B, and on the opposite side of A B take two 



1 If a be the semiaxis, r the radius vector, and v the velocity of P in any 
of an ellipse, a is given 
force being properly assumed. 



2 1 

point of an ellipse, a is given by the relation v 2 — — > the units of velocity and 

r a 



OUTLINES OF ASTRONOMY 563 

points P' and M ', similarly situated with respect to the axis 
with P and M on their side. Then if at P' and M' bodies 
equal to P and M be placed, the forces exerted by M' on 
P' and S will be equal to those exerted by M on P and 
S, and therefore the tangential disturbing force of M' 
on P' exerted in the direction P' Z' (suppose) will equal 
that exerted by M on P in the direction P Z. P' therefore 
(supposing it to revolve in the same direction round S as P) 
will be retarded (or accelerated, as the case may be) by pre- 
cisely the same force by which P is accelerated (or retarded), 
so that the variation in the axis of the respective orbits of 
P and P' will be equal in amount, but contrary in charac- 
ter. Suppose now M's orbit to be circular. Then (if the 
'periodic times of M and P be not commensurate, so that a 
moderate number of revolutions may bring them back to the 
same precise relative positions) it will necessarily happen, 
that in the course of a very great number of revolutions 
of both, bodies, P will have been presented to M on one 
side of the axis, at some one moment, in the same manner 
as at some other moment on the other. Whatever varia- 
tion may have been effected in its axis in the one situation 
will have been reversed in that symmetrically opposite, and 
the ultimate result, on a general average of an infinite num- 
ber of revolutions, will be a complete and exact compen- 
sation of the variations in one direction by those in the 
direction opposite. 

(662.) Suppose, next, P's orbit to be circular. If now 
M's orbit were so also, it is evident that in one complete 
synodic revolution, an exact restoration of the axis to its 
original length would take place, because the tangential 
forces would be symmetrically equal and opposite during 
each alternate quarter revolution. But let M, during a 



561 OUTLINES OF ASTRONOMY 

synodic revolution, have receded somewhat from S, then 
will its disturbing power have become gradually weaker, 
so that, in a synodic revolution the tangential force in 
each quadrant, though reversed in direction being infe- 
rior in power, an exact compensation will not have been 
effected, but there will be left an outstanding uncompen- 
sated portion, the excess of the stronger over the feebler 
effects. But now suppose M to approach by the same gra- 
dations as it before receded. It is clear that this result 
will be reversed; since the uncompensated stronger actions 
will all lie in the opposite direction. Now suppose M's 
orbit to be elliptic. Then during its recess from S or in 
the half revolution from its perihelion to its aphelion, a 
continual uncompensated variation will go on accumulating 
in one direction. But from what has been said, it is clear 
that this will be destroyed, during M's approach to S in 
the other half of its orbit, so that here again, on the aver- 
age of a multitude of revolutions during which P has been 
presented to M in every situation for every distance of M. from 
S, the restoration will be effected. 

(663.) If neither P's nor M's orbit be circular, and if 
moreover the directions of their axes be different, this rea- 
soning, drawn from the symmetry of their relations to each 
other, does not apply, and it becomes necessary to take a 
more general view of the matter. Among the fundamental 
relations of dynamics, relations which presuppose no par- 
ticular law of force like that of gravitation, but which 
express in general terms the results of the action of force 
on matter during time, to produce or change velocity, is one 
usually cited as the ''Principle of the conservation of the 
vis viva, 1 ' which applies directly to the case before us. 
This principle (or rather this theorem) declares that if a 



OUTLINES OF ASTRONOMY 565 

body subjected at every instant of its motion to the action 
of forces directed to fixed centres (no matter how numer- 
ous), and having their intensity dependent only on the dis- 
tances from their respective centres of action, travel from 
one point of space to another, the velocity which it has on 
its arrival at the latter point will differ from that which 
it had on setting out from the former, by a quantity de- 
pending only on the different relative situations of these 
two points in space, without the least reference to the form 
of the curve in which it may have moved in passing from 
one point to the other, whether that curve have been de- 
scribed freely under the simple influence of the central 
forces, or the body have been compelled to glide upon 
it, as a bead upon a smooth wire. Among the forces thus 
acting may be included any constant forces, acting in paral- 
lel directions, which may be regarded as directed to fixed 
centres infinitely distant. It follows from this theorem, 
that, if the body return to the point P from which it set 
out, its velocity of arrival will be the same with that of 
its departure; a conclusion which (for the purpose we have 
in view) sets us free from the necessity of entering into 
any consideration of the laws of the disturbing force, the 
change which its action may have induced in the form of 
the orbit of P, or the successive steps by which velocity 
generated at one point of its intermediate path is destroyed 
at another, by the reversed action of the tangential force. 
Now to apply this theorem to the case in question, let M 
be supposed to retain a fixed position during one whole 
revolution of P. P then is acted on, during that revolu- 
tion, by three forces: 1st, by the central attraction of S 
directed always to S; 2d, by that to M, always directed 
to M; 3d, by a force equal to M's attraction on S; but in 



566 OUTLINES OF ASTRONOMY 

the direction M S, which therefore is a constant force, act- 
ing always in parallel directions. Oa completing its revo- 
lution, then, P 's velocity, and therefore the major axis of 
its orbit, will be found unaltered, at least neglecting that 
excessively minute difference which will result from the 
non-arrival after a revolution at the exact point of its de- 
parture by reason of the perturbations in the orbit produced 
in the interim by the disturbing force, which for the present 
we may neglect. 

(664.) Now suppose M to revolve, and it will appear, 
by a reasoning precisely similar to that of art. 662, that 
whatever uncompensated variation of the velocity arises in 
successive revolutions of P during M's recess from S will 
be destroyed by contrary uncompensated variations arising 
during its approach. Or, more simply and generally thus: 
whatever M's situation may be, for every place which P 
can have, there must exist some other place of P (as P'), in 
which the action of M shall be precisely reversed. Now 
if the periods he incommensurable, in an indefinite number 
of revolutions of both bodies, for every possible combina- 
tion of situations (M, P) there will occur, at some time or 
other, the combination (M, P) which neutralizes the effect 
of the other, when carried to the general account; so that 
ultimately, and when very long periods of time are em- 
braced, a complete compensation will be found to be 
worked out. 

(665.) This supposes, however, that in such long periods 
the orbit of M is not so altered as to render the occurrence 
of the compensating situation (M, P 7 ) impossible. This 
would be the case if M's orbit were to dilate or contract in- 
definitely by a variation in its axis. But the same reason- 
ing which applies to P, applies also to M. P retaining a 



OUTLINES OF ASTRONOMY 567 

I situation, M's velocity, and therefore the axis of its 
orbit, would be exactly restored at the end of a revolution 
of M; so that for every position P M there exists a compen- 
sating position P M'. Thus M's orbit is maintained of the 
same magnitude, and the possibility of the occurrence of the 
compensating situation (M, P') is secured. 

(666.) To demonstrate as a rigorous mathematical truth 
the complete and absolute ultimate compensation of the 
variations in question, it would be requisite to show that 
the minute outstanding changes due to the non-arrivals of P 
and M at the same exact points at the end of each revolution, 
cannot accumulate in the course of infinite ages in one direc- 
tion. Now it will appear in the subsequent part of this 
chapter, that the effect of perturbation on the excentricities 
and apsides of the orbits is to cause the former to undergo 
only periodical variations, and the latter to revolve and take 
up in succession every possible situation. Hence in the 
course of infinite ages, the points of arrival of P and M at 
fixed lines of direction, S P, S M, in successive revolutions, 
though at one time they will approach S, at another will re- 
cede from it, fluctuating to and fro about mean points from 
which they never greatly depart. And if the arrival of 
either of them as P, at a point nearer S, at the end of a com- 
plete revolution, cause an excess of velocity, its arrival at a 
more distant point will cause a deficiency, and thus, as the 
fluctuations of distance to and fro ultimately balance each 
other, so will also the excesses and defects of velocity, 
though in periods of enormous length, being no less than 
that of a complete revolution of P's apsides for the one 
cause of inequality, and of a complete restoration of its ex- 
centricity for the other. 

(667.) The dynamical proposition on which this reasoning 



368 OUTLINES OF ASTRONOMY 

is based is general, and applies equally well to cases wherein 
the forces act in one plane, or are directed to centres any- 
where situated in space. Hence, if we take into considera- 
tion the inclination of P's orbit to that of M, the same rea- 
soning will apply. Only that in this case, upon a complete 
revolution of P, the variation of inclination and the motion 
of the nodes of P's orbit will prevent its returning to a point 
in the exact plane of its original orbit, as that of the excen- 
tricity and perihelion prevent its arrival at the same exact 
distance from S. But since it has been shown that the incli- 
nation fluctuates round a mean state from which it never de- 
parts much, and since the node revolves and makes a com- 
plete circuit, it is obvious that in a complete period of the 
latter the points of arrival of P at the same longitude will 
deviate as often and by the same quantities above as below 
its original point of departure from exact coincidence; and, 
therefore, that on the average of an infinite number of revo- 
lutions, the effect of this cause of non-compensation will also 
be destroyed. 

(668.) It is evident, also, that the dynamical proposition 
in question being general, and applying equally to any num- 
ber of fixed centres, as well as to any distribution of them in 
space, the conclusion would be precisely the same whatever 
be the number of disturbing bodies, only that the periods of 
compensation would become more intricately involved. We 
are, therefore, conducted to this most remarkable and impor- 
tant conclusion, viz. that the major axes of the planetary 
(and lunar) orbits, and, consequently, also their mean mo- 
tions and periodic times, are subject to none but periodical 
changes; that the length of the year, for example, in the 
lapse of infinite ages, has no preponderating tendency either 
to increase or diminution — that the planets will neither re- 



01 ASTRONOMY W» 

cede indefinitely from the sun, nor fall into it, but continue, 
so far as their mutual perturbations at least are concerned, 
to revolve forever in orbits of very nearly the same dimen- 
sions as at present. 

(669.) This theorem (the Magna Gharta of our system), 
the discovery of which is due to Lagrange, is justly regarded 
as the most important, as a single result, of any which have 
hitherto rewarded the researches of mathematicians in this 
application of their science; and it is especially worthy of 
remark, and follows evidently from the view here taken of 
it. that it would not be true but for the influence of the per- 
turbing forces on other elements of the orbit, viz. the peri- 
helion and excentricity, and the inclination and nodes; since 
we have seen that the revolution of the apsides and nodes ? 
and the periodical increase and diminution of the excentrici- 
ties and inclinations, are both essential toward operating 
that final and complete compensation which gives it a char- 
acter of mathematical exactness. We have here an instance 
of a perturbation of one kind operating on a perturbation of 
another to annihilate an effect which would otherwise ac- 
cumulate to the destruction of the system. It must, how- 
ever, be borne in mind, that it is the smallness of the excen- 
tricities of the more influential planets, which gives this 
theorem its practical importance, and distinguishes it from a 
mere barren speculative result. Within the limits of ulti- 
mate restoration, it is this alone which keeps the periodical 
fluctuations of the axis to and fro about a mean value within 
moderate and reasonable limits. Although the earth might 
not fall into the sun, or recede from it beyond the present 
limits of our system, any considerable increase or diminu- 
tion of its mean distance, to the extent, for instance, of a 
tenth of its actual amount, would not fail to subvert the 



570 OUTLINES OF ASTRONOMY 

conditions on which the existence of the present race of ani- 
mated beings depends. Constituted as our system is, how- 
ever, changes to anything like this extent are utterly pre- 
cluded. The greatest departure from the mean value of the 
axis of any planetary orbit yet recognized by theory or ob- 
servation (that of the orbit of Saturn disturbed by Jupiter), 
does not amount to a thousandth part of its length. 2 The 
effects of these fluctuations, however, are very sensible, and 
manifest themselves in alternate accelerations and retarda- 
tions in the angular motions of the disturbed about the cen- 
tral body, which cause it alternately to outrun and to lag 
behind its elliptic place in its orbit, giving rise to what are 
called equations in its motion, some of the chief instances of 
which will be hereafter specified when we come to trace 
more particularly in detail the effects of the tangential force 
in various configurations of the disturbed and disturbing 
bodies, and to explain the consequences of a near approach 
to commensurability in their periodic times. An exact com- 
mensurability in this respect, such, for instance, as would 
bring both planets round to the same configuration in two or 
three revolutions of one of them, would appear at first sight 
to destroy one of the essential elements of our demonstra- 
tion. But even supposing such an exact adjustment to sub- 
sist at any epoch, it could not remain permanent, since by 
a remarkable property of perturbations of this class, which 
geometers have demonstrated, but the reasons of which we 
cannot stop to explain, any change produced on the axis of 
the disturbed planet's orbit is necessarily accompanied by a 
change in the contrary direction in that of the disturbing, so 



2 Greater deviations will probably be found to exist in the orbits of the 
small extra-tropical planets. But these are too insignificant members of our 
system to need special notice in a work of this nature. 



OUTLINES OF ASTRONOMY 



571 



that the periods would recede from commensurability by the 
mere effect of their mutual action. Cases are not wanting 
in the planetary system of a certain approach to commensur- 
ability, and in one very remarkable case (that of Uranus and 
Neptune) of a considerably near one, not near enough, how- 
ever, in the smallest degree to affect the validity of the argu- 
ment, but only to give rise to inequalities of very long 
periods, of which more presently. 8 

(670.) The variation of the length of the axis of the dis- 
turbed orbit is due solely to the action of the tangential dis- 
turbing force. It is otherwise with that of its excentricity 
and of the position of its axis, or, which is the same thing, 
the longitude of its perihelion. Both the normal and tan- 
gential components of the disturbing force affect these ele- 




ments. We shall, however, consider separately the influ- 
ence of each, and, commencing, as the simplest case, with 
that of the tangential force; — let P be the place of the dis- 
turbed planet in its elliptic orbit A P B, whose axis at the 
moment is A S B and focus S. Suppose Y P Z to be a tan- 
gent to this orbit at P. Then, if we suppose A B = 2 «, the 
other focus of the ellipse, H, will be found by making the 



3 41 revolutions of Neptune are nearly equal to 81 of Uranus, giving ris 
to an inequality having 6805 years for its period. 



572 OUTLINES OF ASTRONOMY 

angle ZPH = YPS or YPH = 180°— Y P Z, or S P H 
= 180°— 2 Y P S, and taking PH = 2a-SP. This is evi- 
dent from the nature of the ellipse, in which lines drawn 
from any point to the two foci make equal angles with the 
tangent, and have their sum equal to the major axis. Sup- 
pose, now, the tangential force to act on P and to increase 
its velocity. It will therefore increase the axis, so that the 
new value assumed by a (viz. a') will be greater than a. 
But the tangential force does not alter the angle of tan- 
gency, so that to find the new position (JET) of the upper 
focus, we must measure off along the same line P H, a dis- 
tance P H' (= 2 a' — S P) greater than P H. Do this then, 
and join S H' and produce it. Then will A' B' be the new 
position of the axis, and }SH' the new excentricity. Hence 
we conclude, 1st, that the new position of the perihelion A' 
will deviate from the old one A toward the same side of the 
axis A B on which P is when the tangential force acts to 
increase the velocity, whether P be moving from perihelion 
to aphelion, or the contrary. 2dly, That on the same sup- 
position as to the action of the tangential force, the excen- 
tricity increases when P is between the perihelion and the 
perpendicular to the axis FHG drawn through the upper 
focus, and diminishes when between the aphelion and the 
same perpendicular. 3dly, That for a given change of 
velocity, i.e. for a given value of the tangential force, the 
momentary variation in the place of the perihelion is a maxi- 
mum when P is at F or Gr, from which situation of P to the 
perihelion or aphelion, it decreases to nothing, the perihel- 
ion being stationary when P is at A or B. 4thly, That the 
variation of the excentricity due to this cause is complemen- 
tary in its law of increase and decrease to that of the peri- 
helion, being a maximum for a given tangential force when 



OUTLINES OF ASTRONOMY 



573 



P is at A or B, and vanishing when at Gr or F. And lastly, 
that where the tangential force acts to diminish the velocity, 
all these results are reversed. If the orbit be very nearly 
circular 4 the points F, Gr, will be so situated that, although 
not at opposite extremities of a diameter, the times of de- 
scribing A F, F B, B Gr, and G A will be all equal, and 
each of course one quarter of the whole periodic time of P. 
(671.) Let us now consider the effects of the normal com- 
ponent of the disturbing force upon the same elements. 
The direct effect of this force is to increase or diminish the 
curvature of the orbit at the point P of its action, without 




producing any change on the velocity, so that the length of 
the axis remains unaltered by its action. Now, an increase 
of curvature at P is synonymous with a decrease in the 
angle of tangency SPY when P is approaching toward S, 
and with an increase in that angle when receding from 
S. Suppose the former case, and while P approaches S 
(or is moving from aphelion to perihelion), let the normal 
force act inward or toward the concavity of the ellipse. 
Then will the tangent P Y by the action of that force have 
taken up the position P Y'. To find the corresponding 
position H' taken up by the focus of the orbit so disturbed, 



4 So nearly that the cube of the excentricity may be neglected. 



574 OUTLINES OF ASTRONOMY 

we must make the angle S P H' = 180° — 2 S P Y', or, 
which comes to the same, draw P H" on the side of P H 
opposite to S, making the angle H P H" = twice the angle 
of deflection YPY' and in P H" take PH' = PH. Join- 
ing, then, S H' and producing it, A' S H' M' will be the 
new position of the axis, A' the new perihelion, and \ S H 
the new excentricity. Hence we conclude, 1st, that the 
normal force acting inward, and P moving toward the peri- 
helion, the new direction S A' of the perihelion is in ad- 
vance (with reference to the direction of P's revolution) of 
the old — or the apsides advance — when P is anywhere situ- 
ated between F and A (since when at F the point H' falls 
upon H M between H and M). When P is at F the apsides 
are stationary, but when P is anywhere between M and F 
the apsides retrograde, H' in this case lying on the opposite 
side of the axis. 2dly, That the same directions of the 
normal force and of P's motion being supposed, the ex- 
centricity increases while P moves through the whole semi- 
ellipse from aphelion to perihelion — the rate of its increase 
being a maximum when P is at F, and nothing at the aphe- 
lion and perihelion. 3dly, That these effects are reversed 
in the opposite half of the orbit, A Gr M, in which P passes 
from perihelion to aphelion or recedes from S. 4thly, That 
they are also reversed by a reversal of the direction of the 
normal force, outward, in place of inward. 5thly, That here 
also the variations of the excentricity and perihelion are 
complementary to each other; the one variation being most 
rapid when the other vanishes, and vice versa. 6thly, And 
lastly, that the changes in the situation of the focus H 
produced by the actions of the tangential and normal com- 
ponents of the disturbing force are at right angles to each 
other in every situation of P, and therefore where the 



OUTLINES OF ASTRONOMY 575 

tangential force is most efficacious (in proportion to its 
intensity) in varying either the one or the other of the 
elements in question, the normal is least so, and vice versd. 

(672.) To determine the momentary effect of the whole 
disturbing force then, we have only to resolve it into its 
tangential and normal components, and estimating by these 
principles separately the effects of either constituent on both 
elements, add or subtract the results according as they con- 
spire or oppose each other. Or we may at once make the 
angle H P H" equal to twice the angle of deflection pro- 
duced by the normal force, and lay ofiPH" = PH-f twice 
the variation of a produced in the same moment of time by 
the tangential force, and H" will be the new focus. The 
momentary velocity generated by the tangential force is 
calculable from a knowledge of that force by the ordinary 
principles of dynamics; and from this, the variation of the 
axis is easily derived. 5 The momentary velocity generated 
by the normal force in its own direction is in like manner 
calculable from a knowledge of that force, and dividing this 
by the linear velocity of P at that instant, we deduce the 
angular velocity of the tangent about P, or the momentary 
variation of the angle of tangency SPY, corresponding. 

(673.) The following resume of these several results in a 
tabular form includes every variety of case according as 
Pis approaching to or receding from S; as it is situated 



5 1 2 12 11 

- = v\ and -,= v'-. ' . -, — v 2 — v' 2 = (v 4- v') (v — v') or when in- 

a r a r a a v ' v ' 

a! — a 
finite3imal variations only are considered — — ===2y ( v ' — v ) or a> — «=2a 2 v (y'—v) 

from which it appears that the variation of the axis arising from a given 
variation of velocity is independent of r, or is the same at whatever distance 
from S the change takes place, and that cceieris paribus it is greater for a given 
change of velocity (or for a given tangential force) in the direct ratio of the veloc- 
ity itself. 



576 



OUTLINES OF ASTRONOMY 



in the arc F A Gr of its orbit about the perihelion or in the 
remoter arc Gr M F about the aphelion, as the tangential force 
accelerates or retards the disturbed body, or as the normal 
acts inward or outward with reference to the concavity of 
the orbit. 

EFFECTS OF THE TANGENTIAL DISTURBING FORCE. 



Direction of P's mo- 
tion. 


Situation of P in or- 
bit. 


Action of Tangen- 
tial Force. 


Effect on Elements. 


Approaching S. 


Anywhere 


Accelerating P. 


Apsides recede 


Ditto. 


Ditto 


Retarding P. 


advance 


Receding from S. 


Ditto 


Accelerating P. 


advance 


Ditto 


Ditto 


Retarding P. 


recede 


Indifferent. 


About Aphelion 


Accelerating P. 


Excentr. decreases 


Ditto 


Ditto 


Retarding P. 


increases 


Ditto 


About Perihelion 


Accelerating P. 


increases 


Ditto 


Ditto 


Retarding P. 


decreasesj 



EFFECTS OF THE NORMAL DISTURBING- FORCE. 



Direction of P's mo- 
tion. 


Situation of P in or- 
bit. 


Action of Normal 
Force. 


Effect on Elements. 


Indifferent 


About Aphelion 


Inward 


Apsides recede 


Ditto 


Ditto 


Outward 


advance 


Ditto 


About Perihelion 


Inward 


advance 


Ditto 


Ditto 


Outward 


recede 


Approaching S. 


Anywhere 


Inward 


Excentr. increases 


Ditto 


Ditto 


Outward 


decreases 


Receding from S. 


Ditto 


Inward 


decreases 


Ditto 


Ditto 


Outward 


increases 



(674.) From the momentary changes in the elements of 
the disturbed orbit corresponding to successive situations 
of P and M, to conclude the total amount of change pro- 
duced in any given time is the business of the integral cal- 
culus, and lies far beyond the scope of the present work. 
Without its aid, however, and by general considerations 
of the periodical recurrence of configurations of the same 
character, we have been able to demonstrate many of the 



OUTLINES OF ASTRONOMY 



577 



most interesting conclusions to which geometers have been 
conducted, examples of which have already been given in 
the reasoning by which the permanence of the axes, the 
periodicity of the inclinations, and the revolutions of the 
nodes of the planetary orbits have been demonstrated. We 
shall now proceed to apply similar considerations to the mo- 
tion of the apsides, and the variations of the excentricities. 
To this end we must first trace the changes induced on the 
disturbing forces themselves, with the varying positions of 
the bodies, and here as in treating of the inclinations we 
shall suppose, unless the contrary is expressly indicated, 
both orbits to be very nearly circular, without which limi- 
tation the complication of the subject would become too 
embarrassing for the reader to follow, and defeat the end 
of explanation. 

(675.) On this supposition the directions of S P and S Y, 
the perpendicular on the tangent at P, may be regarded as 
coincident, and the normal and radial disturbing forces 




become nearly identical in quantity, also the tangential and 
transversal, by the near coincidence of the points T and L 
(fig. art. 657). So far then as the intensity of the forces is 
concerned, it will make very little difference in which way 
the forces are resolved, nor will it at all materially affect 



578 OUTLINES OF ASTRONOMY 

our conclusions as to the effects of the normal and tangential 
forces, if in estimating their quantitative values, we take 
advantage of the simplification introduced into their numer- 
ical expression by the neglect of the angle P S Y, Le. by 
the substitution for them of the radial and transversal com- 
ponents. The character of these effects depends (arts. 670, 
671) on the direction in which the forces act, which we shall 
suppose normal and tangential as before, and it is only o^ 
the estimation of their quantitative effects that the error 
induced by the neglect of this angle can fall. Id the lunar 
orbit this angle never exceeds 3° 10', and its influence on 
the quantitative estimation of the acting forces may there- 
fore be safely neglected in a first approximation. Now M N 
being found by the proportion M F:MS'::M S : M N, 
K P (=M 1ST— M P) is also known, and therefore N L=N P. 
sin N P S=N P . sin (A S P+S M P) and L S=P L- 
P S=N P. cos N P S-P S=N P . cos (A S P+S M P) 
— S P become known, which express respectively the tan- 
gential and normal forces on the same scale that S M rep- 
resents M's attraction on S. 6 Suppose P to revolve in the 
direction EADB. Then, by drawing the figure in vari- 
ous situations of P throughout the whole circle, the reader 
will easily satisfy himself — 1st. That the tangential force 
accelerates P, as it moves from E toward A, and from D 



6 MS = R; SP-r; MP-/; ASP = <9; AMP=J4; MN-t; NP = 
~~- ={R— /) ( 1 +-y+-p) ; whence we have NL=(R- /). sin(6-f-M) 

(l+y + ^j;LS = (R-/).cos(6>-i-M). (l + j+jj-r. When R and 

/, owing to the great distance of M, are nearly equal, we have R— /==• 

R 
FY, -j «=«1 nearly, and the angle M may be neglected ; so that we have N F 

-3PV. 



OUTLINES OF ASTRONOMY 579 

toward B, but retards it as it passes from A to D, and from 
B to i'j. %dlj. That the tangential force vanishes at the four 
points A, D, E, B, and attains a maximum at some inter- 
mediate points. 3dly. That the normal force is directed 
outward at the syzygies A, B, and inward at the points 
D, E, at which points respectively its outward and inward 
intensities attain their maxima. Lastly, that this force 
vanishes at points intermediate between A D, D B, B E, 
and E A, which points, when M is considerably remote, 
are situated nearer to the quadrature than the syzygies. 

(676.) In the lunar theory, to which we shall now pro- 
ceed to apply these principles, both the geometrical repre- 
sentation and the algebraic expression of the disturbing 
forces admit of great simplification. Owing to the great 
distance of the sun M, at whose centre the radius of the 




moon's orbit never subtends an angle of more than about 
8', N P may be regarded as parallel to A B. And D S E 
becomes a straight line coincident with the line of quadra- 
tures, so that V P becomes the cosine of ASP, to radius 
S P, and N L=N P . sin ASP; L P = N P . cos A S P. 
Moreover, in this case (see the note on the last article) 
$ P = 3 P V = 3 S P . cos A S P; and consequently N L 

5=3 S P . cos A St P - sin A 3 P=| S P . sin 2 A S P, and 

Astronomy — Yol. XX — 6 



580 OUTLINES OF ASTRONOMY 

LS=S P(3 . cos ASP 2 — 1)=J SP(l+3 . cos 2 A S P) 
which vanishes when cos A S P 2 =3, or at 64° 14' from the 
syzygy. Suppose through every point of P's orbit there 
be drawn S Q=3 S P . cos A S P 2 , then will Q trace out 
a certain looped oval, as in the figure, cutting the orbit in 
four points 64° 14' from A and B respectively, and P Q will 
always represent in quantity and direction the normal force 
acting at P. 

(677.) It is important to remark here, because upon this 
the whole lunar theory and especially that of the motion 
of the apsides hinges, that all the acting disturbing forces, 
at equal angles of elongation A S P of the moon from the 
sun, are cceteris paribus proportional to S P, the moon's 
distance from the earth, and are therefore greater when 



+ + 




the moon is near its apogee than when near its perigee; 
the extreme proportion being that of about 28 : 25. This 
premised, let us first consider the effect of the normal 
force in displacing the lunar apsides. This we shall best 
be enabled to do by examining separately those cases in 
which the effects are most strongly contrasted, viz. when 
the major axis of the moon's orbit is directed toward the 
sun, and when at right angles to that direction. First, 
then, let the line of apsides be directed to the sun as in 



OUTLINES OF ASTRONOMY 581 

the annexed figure, where A is the perigee, and take the 
arcs A a, A b, B c, B d each = 6-±° 14'. Then while P is 
between a and b the normal force acting outward, and the 
moon being near its perigee, by art. 671, the apsides will 
rece<te, but when between c and d, the force there acting 
outward, but the moon being near its apogee, they will 
advance. The rapidity of these movements will be respect- 
ively at its maxima at A and B, not only because the dis- 
turbing forces are then most intense, but also because (see 
art. 671) they act most advantageously at those points to 
displace the axis. Proceeding from A and B toward the 
neutral points abed the rapidity of their recess and ad- 
vance diminishes, and is nothing (or the apsides are sta- 
tionary) when P is at either of these points. From b to 
D, or rather to a point some little beyond D (art. 671) acts 
inward, and the moon is still near perigee, so that in this 
arc of the orbit the apsides advance. But the rate of ad- 
vance is feeble, because in the early part of that arc the 
normal force is small, and as P approaches D and the force 
gains power, it acts disadvantageously to move the axis, 
its effect vanishing altogether when it arrives beyond D at 
the extremity of the perpendicular to the upper focus of 
the lunar ellipse. Thence up to c this feeble advance is 
reversed and converted into a recess, the force still acting 
inward, but the moon now being near its apogee. And so 
also for the arcs d E, E a. In the figure these changes are 

indicated by -\~\- for rapid advance,- for rapid recess, 

-f and — for feeble advance and recess, and for the sta- 
tionary points. Now if the forces were equal on the sides 
of + and — it is evident that there would be an exact coun- 
terbalance of advance and recess on the average of a whole 
revolution. But this is not the case. The force in apogee 



582 OUTLINES OF ASTRONOMY 

is greater than that in perigee in the proportion of 28 : 25, 
while in the quadratures about D and E they are equal. 
Therefore, while the feeble movements + aD -d — in the 
neighborhood of these points destroy each other almost 
exactly, there will necessarily remain a considerable bal- 
ance in favor of advance, in this situation of the line 
of apsides. 

(678.) Next, suppose the apogee to lie at A, and the 
perigee at B. In this case it is evident that, so far as 
the direction of the motions of the apsides is concerned, all 
the conclusions of the foregoing reasoning will be reversed 
by the substitution of the word perigee for apogee, and vice 
versa; and all the signs in the figure referred to will be 
changed. But now the most powerful forces act on the 
side of A, that is to say, still on the side of advance, 
this condition also being reversed. In either situation of 
the orbit, then, the apsides advance. 

(679.) (Case 3.) Suppose, now, the major axis to have 
the situation D E, aud the perigee to be on the side of D. 
Here, in the arc b c of P's motion the normal force acts 
inward, and the moon is near perigee, consequently the 
apsides advance, but with a moderate rapidity, the maxi- 
mum of the inward normal force being only half that of 
the outward. In the arcs A b and c B the moon is still 
near perigee, and the force acts outward, but though 
powerfully toward A and B, yet at a constantly increas- 
ing disadvantage (art. 671). Therefore in these arcs the 
apsides recede, but moderately. In a A and B d (being 
toward apogee) they again advance, still with a moderate 
velocity. Lastly, throughout the arc d a, being about apogee 
with an inward force, they recede. Here as before, if the 
perigee and apogee forces were equal, the advance and re- 



OUTLINES OF ASTRONOMY 583 

would counterbalance; but as in fact the apogee forces 
preponderate, there will be a balance on the entire revolu- 
tion in favor of recess. The same reasoning of course holds 
good if the perigee be toward E. But now, between these 
cases and those in the foregoing articles, there is this dif- 
ference, viz. that in this the dominant effect results from 
the inward action of the normal force in quadratures, while 
in the others it results from its outward, and doubly power- 
ful action in syzygies. The recess of the apsides in their 
quadratures arising from the action of the normal force 
will therefore be less than their advance in their syzygies; 
and not only on this account, but also because of the much 
less extent of the arcs b c and d a on which the balance is 
mainly struck in this case, than of a b and c d, the corre- 
sponding most influential arcs in the other. 

(680.) In intermediate situations of the line of apsides, 
the effect will be intermediate, and there will of course be 
a situation of them in which on an average of a whole 
revolution, they are stationary. This situation it is easy 
to see will be nearer to the line of quadratures than of 
syzygies, and the preponderance of advance will be main- 
tained over a much more considerable arc than that of 
recess, among the possible situations which they can hold. 
On every account, therefore, the action of the normal force 
causes the lunar apsides to progress in a complete revolu- 
tion of M or in a synodical year, during which the motion 
of the sun round the earth (as we consider the earth at 
rest) brings the line of syzygies into all situations with 
respect to that of apsides. 

(681.) Let us next consider the action of the tangential 
force. And as before (Case 1), supposing the perigee of 
the moon at A, and the direction of her revolution to be 



584 OUTLINES OF ASTRONOMY 

ADBE, the tangential force retards her motion through 
the quadrant A D, in which she recedes from S, therefore 
by art. 670, the apsides recede. Through D B the force 
accelerates, while the moon still recedes, therefore they ad- 
vance. Through B E the force retards, and the moon ap- 
proaches, therefore they continue to advance, and finally 
throughout the quadrant E A the force accelerates and the 
moon approaches, therefore they recede. In virtue there- 
fore of this force, the apsides recede, during the descrip- 
tion of the arc E A D, and advance during D B E, but 
the force being in this case as in that of the normal force 
more powerful at apogee, the latter will preponderate, and 
the apsides will advance on an average of a whole revolution. 

(682.) (Case 2.) The perigee being toward B, we have 
to substitute in the foregoing reasoning approach to S, for 
recess from it, and vice versa, the accelerations and retarda- 
tions remaining as before. Therefore the results, as far as 
direction is concerned, will be reversed in each quadrant, 
the apsides advance during E A D and recede during 
D B E. But the situation of the apogee being also re- 
versed, the predominance remains on the side of E A D, 
that is, of advance. 

(683.) (Case 3.) Apsides in quadratures, perigee near 
D. — Over quadrant A D, approach and retardation, there- 
fore advance of apsides. Over D B recess and acceleration, 
therefore again advance; over B E recess and retardation 
with recess of apsides, and lastly over E A approach and 
acceleration, producing their continued recess. Total re- 
sult: advance during the half revolution A D B, and recess 
during B E A, the acting forces being more powerful in the 
latter, whence of course a preponderant recess. The same 
result when the perigee is at E. 



OUTLINES OF ASTRONOMY 585 

(684.) So far the analogy of reasoning between the action 
of the tangential and normal forces is perfect. But from 
this point they diverge. . It is not here as before. The 
recess of the apsides in quadratures does not now arise 
from the predominance of feeble over feebler forces, while 
that in syzygies results from that of powerful over powerful 
ones. The maximum accelerating action of the tangential 
force is equal to its maximum retarding, while the inward 
action of the normal at its maximum is only half the maxi- 
mum of its outward. Neither is there that difference in the 
extent of the arcs over which the balance is struck in this, 
as in the other case, the action of the tangential force be- 
ing inward and outward alternately over equal arcs, each 
a complete quadrant. Whereas, therefore, in tracing the 
action of the normal force, we found reason to conclude 
it much more effective to produce progress of the apsides 
in their syzygy, than in their quadrature situations, we 
can draw no such conclusion in that of the tangential 
forces: there being, as regards that force, a complete sym- 
metry in the four quadrants, while in regard of the normal 
force the symmetry is only a half-symmetry having relation 
to two semicircles. 

(685.) Taking the average of many revolutions of the sun 
about the earth, in which it shall present itself in every pos- 
sible variety of situations to the line of apsides, we see that 
the effect of the normal force is to produce a rapid advance 
in the syzygy of the apsides, and a less rapid recess in their 
quadrature, and on the whole, therefore, a moderately rapid 
general advance, while that of the tangential is to produce 
an equally rapid advance in syzygy, and recess in quadra- 
ture. Directly, therefore, the tangential force would appear 
to have no ultimate influence in causing either increase or 



686 OUTLINES OF ASTRONOMY 

diminution in the mean motion of the apsides resulting from 
the action of the normal force. It does so, however, indi- 
rectly, conspiring in that respect with, and greatly increas- 
ing, an indirect action of the normal force in a manner which 
we shall now proceed to explain. 

(686.) The sun moving uniformly, or nearly so, in the 
same direction as P, the line of apsides when in or near the 
syzygy, in advancing follows the sun, and therefore remains 
materially longer in the neighborhood of syzygy than if it 
rested. On the other hand, when the apsides are in quad- 
rature they recede, and moving therefore contrary to the 
sun's motion, remain a shorter time in that neighborhood, 
than if they rested. Thus the advance, already preponder- 
ant, is made to preponderate more by its longer continuance, 
and the recess, already deficient, is rendered still more so by 
the shortening of its duration. 7 Whatever cause, then, in- 
creases directly the rapidity of both advance and recess, 
though it may do both equally, aids in this indirect process, 
and it is thus that the tangential force becomes effective 
through the medium of the progress already produced, in 
doing and aiding the normal force to do that which alone it 
would be unable to effect. Thus we have perturbation ex- 
aggerating perturbation, and thus we see what is meant by 
geometers, when they declare that a considerable part of the 
motion of the lunar apsides is due to the square of the dis- 
turbing force, or, in other words, arises out of a second ap- 
proximation in which the influence of the first in altering 
the data of the problem is taken into account. 

(687.) The curious and complicated effect of perturba- 
tion, described in the last article, has given more trouble to 
geometers than any other part of the lunar theory. Newton 

7 Kewt-bn. !Pxinc. i. Go. Cor. 8. 



OUTLINES OF ASTRONOMY 587 

himself had succeeded in tracing that part of the motion of 
the apogee which is due to the direct action of the radial 
force; but finding the amount only half what observation 
assigns, he appears to have abandoned the subject in despair. 
Nor, when resumed by his successors, did the inquiry, for a 
very long period, assume a more promising aspect. On the 
contrary, Newton's result appeared to be even minutely 
verified, and the elaborate investigations which were lav- 
ished upon the subject without success began to excite 
strong doubts whether this feature of the lunar motions 
could be explained at all by the Newtonian law of gravita- 
tion. The doubt was removed, however, almost in the in- 
stant of its origin, by the same geometer, Clairaut, who first 
gave it currency, and who gloriously repaired the error of 
his momentary hesitation, by demonstrating the exact coin- 
cidence between theory and observation, when the effect of 
the tangential force is properly taken into the account. The 
lunar apogee circulates in 3232 d -575343, or about 9£ years. 
(688.) Let us now proceed to investigate the influence of 
the disturbing forces so resolved on the excentricity of the 
lunar orbit, and the foregoing articles having sufficiently 
familiarized the reader with our mode of following out the 
changes in different situations of the °^*-*~7r°*****S) 

orbit, we shall take at once a more / 
general situation, and suppose the line / 
of apsides in any position with respect I b 
to the sun, such as Z Y, the perigee 
being at Z, a point between the lower °^ 
syzygy and the quadrature next follow- 
ing it, the direction of P's motion as all along supposed 
being A D B E. Then (commencing with the normal 
force) the momentary change of excentricity will vanish 




588 OUTLINES OF ASTRONOMY 

at a, b, c, d, by the vanishing of that force, and at Z 
and Y by the effect of situation in the orbit annulling 
its action (art. 671). In the arcs Z b and Y d therefore 
the change of excentricity will be small, the acting force 
nowhere attaining either a great magnitude or an ad- 
vantageous situation within their limits. And the force 
within these two arcs having the same character as to in- 
ward and outward, but being oppositely influential by rea- 
son of the approach of P to S in one of them and its recess 
in the other, it is evident that, so far as these arcs are con- 
cerned, a very near compensation of effects will take place, 
and though the apogeal arc Y d will be somewhat more 
influential, this will tell for little upon the average of a 
revolution. 

(689.) The arcs h D c and d E a are each much less than 
a quadrant in extent, and the force acting inward through- 
out them (which at its maximum in D and E is only half the 
outward force at A, B) degrades very rapidly in intensity 
toward either syzygy (see art. 676). Hence whether Z be 
between b c or h A, the effects of the force in these arcs will 
not produce very extensive changes on the excentricity, and 
the changes which it does produce will (for the reason al- 
ready given) be opposed to each other. Although, them 
the arc a d be further from perigee than b c, and therefore 
the force in it is greater, yet the predominance of effect here 
will not be very marked, and will moreover be partially 
neutralized by the small predominance of an opposite char- 
acter in Yd over Z b. On the other hand, the arcs a Z, c Y 
are both larger in extent than either of the others, and the 
seats of action of forces doubly powerful. Their influence, 
therefore, will be of most importance, and their preponder- 
ance one over the other (being opposite in their tendencies) 



OUTLINES OF ASTRONOMY 589 

will decide the question whether on an average of the revo- 
lution, the excentricity shall increase or diminish. It is 
clear that the decision must be in favor of c Y, the apogeal 
arc, and, since in this the force is outward and the moon 
receding from the earth, an increase of the excentricity will 
arise from its influence. A similar reasoning will, evi- 
dently, lead to the same conclusion were the apogee and 
perigee to change places, for the directions of P's motion as 
to approach and recess to S will be indeed reversed, but at 
the same time the dominant forces will have changed sides, 
and the arc a A Z will now give the character to the result. 
But when Z lies between A and E, as the reader may easily 
satisfy himself, the case will be altogether different, and the 
reverse conclusion will obtain. Hence the changes of ex- 
centricity emergent on the average of single revolutions from 
the action of the normal force will be as represented by the 
signs -+- and — in the figure above annexed. 

(690.) Let us next consider the effect of the tangential 
force. This retards P in the quadrants A D, B E, and ac- 
celerates it in the alternate ones. In the whole quadrant 
A D, therefore, the effect is of one character, the perigee 
being less than 90° from every point p 

in it, and in the whole quadrant B E 
it is of the opposite, the apogee being 
so situated (art. 670). Moreover, in 




the middle of each quadrant, the tan- 
gential force is at its maximum. 
Now, in the other quadrants, E A 
and D B, the change from perigeal to apogeal vicinity takes 
place, and the tangential force, however powerful, has its 
effect annulled by situation (art. 670), and this happens 
more or less nearly about the points where the force is a 



690 OUTLINES OF ASTRONOMY 

maximum. These quadrants, then, are far le*?. influential 
on the total result, so that the character of that result will 
be decided by the predominance of one or other of the 
former quadrants, and will lean to that which has the 
apogee in it. Now in the quadrant B E the force retards 
the moon and the moon is in apogee. Therefore the excen- 
tricity increases. In this situation therefore of the apogee, 
such is the average result of a complete revolution of the 
moon. Here again also if the perigee and apogee change 
places, so will also the character of all the partial influences, 
arc for arc. But the quadrant AD will now preponderate 
instead of D E, so that under this double reversal of condi- 
tions the result will be identical. Lastly, if the line of ap- 
sides be in A E, B D, it may be shown in like manner that 
the excentricity will diminish on the average of a revolution. 
(691.) Thus it appears, that in varying the excentricity, 
precisely as in moving the line of apsides, the direct effect 
of the tangential force conspires with that of the normal, 
and tends to increase the extent of the deviations to and fro 
on either side of a mean value which the varying situation 
of the sun with respect to the line of apsides gives rise to, 
having for their period of restoration a synodical revolution 
of the sun and apse. Supposing the sun and apsis to start 
together, the sun of course will outrun the apsis (whose 
period is nine .years), and in the lapse of about (J -j- 32) part 
of a year will have gained on it 90°, during all which inter- 
val the apse will have been in the quadrant A E of our 
figure, and the excentricity continually decreasing. The 
decrease will then cease, but the excentricity itself will be 
a minimum, the sun being now at right angles to the line of 
apsides. Thence it will increase to a maximum when the 
sun has gained another 90°, and again attained the line of 



OUTLINES OF ASTRONOMY 59l 

apsides, and so on alternately. The actual eifect on the 
numerical value of the lunar excentricity is very consider- 
able, the greatest and least excentricities being in the ratio 
of 3 to 2. 8 

(692.) The motion of the apsides of the lunar orbit may 
be illustrated by a very pretty mechanical experiment, 
which is otherwise instructive in giving an idea of the 
mode in which orbitual motion is carried on under the action 
of central forces variable according to the situation of the 
revolving body. Let a leaden weight be suspended by a 
brass or iron wire to a hook in the under side of a firm 
beam, so as to allow of its free motion on all sides of the 
vertical, and so that when in a state of rest it shall just clear 
the floor of the room, or a table placed ten or twelve feet 
beneath the hook. The point of support should be well 
secured from wagging to and fro by the oscillation of the 
weight, which should be sufficient to keep the wire as tightly 
stretched as it will bear, with the certainty of not breaking. 
Now, let a very small motion be communicated to the 
weight, not by merely withdrawing it from the vertical and 
letting it fall, but by giving it a slight impulse sidewise. 
It will be seen to describe a regular ellipse about the point 
of rest as its centre. If the weight be heavy, and carry 
attached to it a pencil, whose point lies exactly in the direc- 
tion of the string, the ellipse may be transferred to paper 
lightly stretched and gently pressed against it. In these 
circumstances, the situation of the major and minor axes 
of the ellipse will remain for a long time very nearly the 
same, though the resistance of the air and the stiffness of 
the wire will gradually diminish its dimensions and excen- 
tricity. But if the impulse communicated to the weight be 

8 Airy, Gravitation, p. 106. 



592 OUTLINES OF ASTRONOMY 

considerable, so as to carry it out to a great angle (15° or 
20° from the vertical), this permanence of situation of the 
ellipse will no longer subsist. Its axis will be seen to shift 
its position at every revolution of the weight, advancing in 
the same direction with the weight's motion, by a uniform 
and regular progression, which at length will entirely re- 
verse its situation, bringing the direction of the longest 
excursions to coincide with that in which the shortest were 
previously made; and so on, round the whole circle; and, 
in a word, imitating to the eye, very completely, the motion 
of the apsides of the moon's orbit. 

(693.) Now, if we inquire into the cause of this pro- 
gression of the apsides, it will not be difficult of detection. 
WheD a weight is suspended by a wire, and drawn aside 
from the vertical, it is urged to the lowest point (or rather 
in a direction at every instant perpendicular to the wire) by 
a force which varies as the sine of the deviation of the wire 
from the perpendicular. Now, the sines of very small arcs 
are nearly in the proportion of the arcs themselves; and the 
more nearly, as the arcs are smaller. If, therefore, the de- 
viations from the vertical be so small that we may neglect 
the curvature of the spherical surface in which the weight 
moves, and regard the curve described as coincident with 
its projection on a horizontal plane, it will be then moving 
under the same circumstances as if it were a revolving body 
attracted to a centre by a force varying directly as the 
distance; and, in this case, the curve described would be 
an ellipse, having its centre of attraction not in the focus, 
but in the centre, 9 and the apsides of this ellipse would re- 
main fixed. But if the excursions of the weight from the 
vertical be considerable, the force urging it toward the 

9 Newton, Princip. i. 47. 



OUTLINES OF ASTRONOMY 593 

centre will deviate in its law from the simple ratio of 
distances] being as the sine, while the distances are as 
arc. Now the sine, though it continues to increase 
as the arc increases, yet does not increase so fast. So soon 
as the arc has any sensible extent, the sine begins to fail 
somewhat short of the magnitude which an exact numeri- 
cal proportionality would require; and therefore the force 
urging the weight toward its centre or point of rest at great 
distances falls, in like proportion, somewhat short of that 
which would keep the body in its precise elliptic orbit. It 
will no longer, therefore, have, at those greater distances, 
the same command over the weight, in proportion to its 
speed, which would enable it to deflect it from its rec- 
tilinear tangential course into an ellipse. The true path 
which it describes will be less curved in the remoter parts 
than is consistent with the elliptic figure, as in the annexed 
cut; and, therefore, it will not so soon have its motion 
brought to be again at right angles to ^^.- * 

the radius. It will require a longer f " V"""^ 
continued action of the central force to J! \ 

do this; and before it is accomplished. *>,- -———-—' » 
more than a quadrant of its revolution '-x^ 
must be passed over in angular motion ~ ^*=--= — ~L-- 

round the centre. But this is only stating at length, and in 
a more circuitous manner, that fact which is more briefly 
and summarily expressed by saying that the apsides of its 
orbit are progressive. Nothing beyond a familiar illustration 
is of course intended in what is above said. The case is not 
an exact parallel with that of the lunar orbit, the disturbing 
force being simply radial, whereas in the lunar orbit a trans- 
versal force is also concerned, and even were it otherwise, 
only a confused and indistinct view of apsidal motion can 



594 OUTLINES OF ASTRONOMY 

be obtained from this kind of consideration of the curvature 
of the disturbed path. If we would obtain a clear one, the 
two foci of the instantaneous ellipse must be found from 
the laws of elliptic motion performed under the influence of 
a force directly as the distance, and the radial disturbing 
force being decomposed into its tangential and normal com- 
ponents, the momentary influence of either in altering their 
positions and consequently the directions and lengths of the 
axis of the ellipse must be ascertained. The student will 
find it neither a difficult nor an uninstructive exercise to 
work out the case from these principles, which we cannot 
afford the space to do. 

(694.) The theory of the motion of the planetary apsides 
and the variation of their excentricities is in one point of 
view much more simple, but in another much more compli- 
cated than that of the lunar. It is simpler, because owing 
to the exceeding minuteness of the changes operated in 
the course of a single revolution, the angular position of 
the bodies with respect to the line of apsides is very little 
altered by the motion of the apsides themselves. The line 
of apsides neither follows up the motion of the disturbing 
body in its state of advance, nor vice versa, in any degree 
capable of prolonging materially their advancing or shorten- 
ing materially their receding phase. Hence no second ap- 
proximation of the kind explained in art. 686, by which the 
motion of the lunar apsides is so powerfully modified as to 
be actually doubled in amount, is at all required in the 
planetary theory. On the other hand, the latter theory 
is rendered more complicated than the former, at least in 
the cases of planets whose periodic times are to each other 
in a ratio much less than 13 to 1, by the consideration that 
the disturbing body shifts its position with respect to the 



OUTLINES OF ASTRONOMY 595 

of apsides by a much greater angular quantity in a 
revolution of the disturbed body than in the case of the 
moon. In that case we were at liberty to suppose (for 
the sake of explanation), without any very egregious error, 
that the sun held nearly a fixed position during a single 
lunation. But in the case of planets whose times of revo- 
lution are in a much lower ratio this cannot be permitted. 
In the case of Jupiter disturbed by Saturn for example, in 
one sidereal revolution of Jupiter, Saturn has advanced 
in its orbit with respect to the line of apsides of Jupiter by 
more, than 140°, a change of direction which entirely alters 
the conditions under which the disturbing forces act. And 
in the case of an exterior disturbed by an interior planet, 
the situation of the latter with respect to the line of the 
apsiles varies even more rapidly than the situation of 
the exterior or disturbed planet with respect to the central 
body. To such cases then the reasoning which we have 
applied to the lunar perturbations becomes totally inappli- 
cable; and when we take into consideration also the excen- 
tricity of the orbit of the disturbing body, which in the 
most important cases is exceedingly influential, the subject 
becomes far too complicated for verbal explanation, and can 
only be successfully followed out with the help of algebraic 
expression and the application of the integral calculus, To 
Mercury, Venus, and the earth indeed, as disturbed by 
Jupiter," and planets superior to Jupiter, this objection 
to the reasoning in question does not apply; and in each 
of these cases therefore we are entitled to conclude that 
the apsides are kept in a state of progression by the action 
of all the larger planets of our system. Under certain 
conditions of distance, excentricity, and relative situation 
of the axes of the orbits of the disturbed and disturbing 



596 OUTLINES OF ASTRONOMY 

planets, it is perfectly possible that the reverse may hap- 
pen, an instance of which is afforded by Venus, whose 
apsides recede under the combined action of the earth and 
Mercury more rapidly than they advance under the joint 
actions of all the other . planets. Nay, it is even possible 
under certain conditions that the line of apsides of the dis- 
turbed planet, instead of revolving always in one direction, 
may librate to and fro within assignable limits, and in a 
definite and regularly recurring period of time. 

(695.) Under any conditions, however, as to these par- 
ticulars, the view we have above taken of the subject, 
enables us to assign at every instant, and in every con- 
figuration of the two planets, the momentary effect of each 
upon the perihelion and excentricity of the other. In the 
simplest case, that in which the two orbits are so nearly 
circular, that the relative situation of their perihelia shall 
produce no appreciable difference in the intensities of the 
disturbing forces, it is very easy to show that whatever 
temporary oscillations to and fro in the positions of the 
line of apsides, and whatever temporary increase and 
diminution in the excentricity of either planet may take 
place, the final effect on the average of a great multitude 
of revolutions, presenting them to each other in all possible 
configurations, must be nil, for both elements. 

(696.) To show this, all that is necessary is to cast our 
eyes on the synoptic table in art. 673. If M, the disturbing 
body, be supposed to be successively placed in two dia- 
metrically opposite situations in its orbit, the aphelion of 
P will stand related to M in one of these situations precisely 
as its perihelion in the other. Now the orbits being so 
nearly circles as supposed, the distribution of the disturb- 
ing forces, whether normal or tangential, is symmetrical 



OUTLINES OF ASTRONOJMY 597 

relative to their common diameter passing through M, or 
to the line of syzygies. Hence it follows that the half of P's 
orbit "about perihelion" (art. 673) will stand related to all 
the acting forces in the one situation of M, precisely as the 
half "about aphelion" does in the other: and also, that 
the half of the orbit in which P "approaches S," stands 
related to them in the one situation precisely as the half in 
which it "recedes from S" in the other. Whether as re- 
gards, therefore, the normal or tangential force, the condi- 
tions of advance or recess of apsides, and of increase or 
diminution of excentricities, are reversed in the two sup- 
posed cases. Hence it appears that whatever situation be 
assigned to M, and whatever influence it may exert on P in 
that situation, that influence will be annihilated in situations 
of M and P, diametrically opposite to those supposed, and 
thus, on a general average, the effect on both apsides and 
excentricities is reduced to nothing. 

(697.) If the orbits, however, be excentric, the symmetry 
above insisted on in the distribution of the forces does not 
exist. But, in the first place, it is evident that if the excen- 
tricities be moderate (as in the planetary orbits), by far the 
larger part of the effects of the disturbing forces destroys 
itself in the manner described in the last article, and that it 
is only a residual portion, viz. that which arises from the 
greater proximity of the orbits at one place than at another, 
which can tend to produce permanent or secular effects. 
The precise estimation of these effects is too complicated an 
affair for us to enter upon ; but we may at least give some idea 
of the process by which they are produced, and the order in 
which they arise. In so doing, it is necessary to distinguish 
between the effects of the normal and tangential forces. 
The effects of the former are greatest at the point of con- 



598 OUTLINES OF ASTRONOMY 

junction of the planets, because the normal force itself is 
there always at its maximum; and although, where the con- 
junction takes place at 90° from the line of apsides, its effect 
to move the apsides is nullified by situation, and when in 
that line its effect on the excentricities is similarly nullified, 
yet, in the situations rectangular to these, it acts to its great- 
est advantage. On the other hand, the tangential force van- 
ishes at conjunction, whatever be the place of conjunction 
with respect to the line of apsides, and where it is at its 
maximum its effect is still liable to be annulled by situa- 
tion. Thus it appears that the normal force is most influen- 
tial, and mainly determines the character of the general 
effect. It is, therefore, at conjunction that the most influ- 
ential effect is produced, and therefore, on the long average, 
those conjunctions which happen about the place where the 
orbits are nearest will determine the general character of 
the effect. ISTow, the nearest points of approach of two 
ellipses which have a common focus may be very variously 
situated with respect to the perihelion of either. It may be 
at the perihelion or the aphelion of the disturbed orbit, or 
in any intermediate position. Suppose it to be at the peri- 
helion. Then, if the disturbed orbit be interior to the dis- 
turbing, the force acts outward, and therefore the apsides 
recede: if exterior, the force acts inward, and they advance. 
In neither case does the excentricity change. If the con- 
junction take place at the aphelion of the disturbed orbit, 
the effects will be reversed: if intermediate, the apsides will 
be less, and the excentricity more affected. 

(698.) Supposing only two planets, this process would go 
on till the apsides and excentricities had so far changed as 
to alter the point of nearest approach of the orbits so as 
either to accelerate or retard and perhaps reverse the motion 



OUTLINES OF ASTRONOMY 599 

of the apsides, and give to the variation of the excentricity 
a corresponding periodical character. But there are many 
planets all disturbing one another. And this gives rise to 
variations in the points of nearest approach of all the orbits 
taken two and two together, of a very complex nature. 

(699.) It cannot fail to have been remarked, by any one 
who has followed attentively the above reasonings, that a 
close analogy subsists between two sets of relations; viz. 
that between the inclinations and nodes on the one hand, 
and between the excentricity and apsides on the other. In 
fact, the strict geometrical theories of the two cases present 
a close analogy, and lead to final results of the very same 
nature. What the variation of excentricity is to the motion 
of the perihelion, the change of inclination is to the motion 
of the node. In either case, the period of the one is also the 
period of the other; and while the perihelia describe consid- 
erable angles by an oscillatory motion to and fro, or circu- 
late in immense periods of time round the entire circle, the 
excentricities increase and decrease by comparatively small 
changes, and are at length restored to their original magni- 
tudes. In the lunar orbit, as the rapid rotation of the nodes 
prevents the change of inclination from accumulating to any 
material amount, so the still more rapid revolution of its 
apogee effects a speedy compensation in the fluctuations of 
its excentricity, and never suffers them to go to any mate- 
rial extent; while the same causes, by presenting in quick 
succession the lunar orbit in every possible situation to all 
the disturbing forces, whether of the sun, the planets, or the 
protuberant matter at the earth's equator, prevent any secu- 
lar accumulation of small changes, by which, in the lapse of 
ages, its elliptic] ty might be materially increased or dimin- 
ished. Accordingly, observation shows the mean excentric- 



600 OUTLINES OF ASTRONOMY 

ity of the moon's orbit to be the same now as in the earliest 
ages of astronomy. 

(700.) The movements of the perihelia, and variations of 
excentricity of the planetary orbits, are interlaced and com- 
plicated together in the same manner and nearly by the 
same laws as the variations of their nodes and inclinations. 
Each acts upon every other, and every such mutual action 
generates its own peculiar period of circulation or compen- 
sation, and every such period, in pursuance of the principle 
of art. 650, is thence propagated throughout the system. 
Thus arise cycles upon cycles, of whose compound duration 
some notion may be formed, when we consider what is the 
length of one such period in the case of the two principal 
planets — Jupiter and Saturn. Neglecting the action of the 
rest, the effect of their mutual attraction would be to pro- 
duce a secular variation in the excentricity of Saturn's orbit, 
from 0-08409, its maximum, to 0-01345, its minimum value: 
while that of Jupiter would vary between the narrow limits, 
0*06036 and 0*02606: the greatest excentricity of Jupiter 
corresponding to the least of Saturn, and vice versa. The 
period in which these changes are gone through would be 
70414 years. After this example, it will be easily conceived 
that many millions of years will require to elapse before a 
complete fulfilment of the joint cycle which shall restore the 
whole system to its original state as far as the excentricities 
of its orbits are concerned. 

(701.) The place of the perihelion of a planet's orbit is 
of little consequence to its well-being; but its excentricity is 
most important, as upon this (the axes of the orbits being 
permanent) depends the mean temperature of its surface, 
and the extreme variations to which its seasons may be lia- 
ble. For it may be easily shown that the mean annual 



OUTLINES OF ASTRONOMY 601 

amount of light and heat received by a planet from the sun 
is, cceteris paribus, as the minor axis of the ellipse described 
by it. Any variation, therefore, in the excentricity, by 
changing the minor axis will alter the mean temperature of 
the surface. How such a change will also influence the ex- 
tremes of temperature appears from art. 368 et sea. Now it 
may naturally be inquired whether (in the vast cycle above 
spoken of, in which, at some period or other, conspiring 
changes may accumulate on the orbit of one planet from 
several quarters) it may not happen that the excentricity 
of any one planet — as the earth — may become exorbitantly 
great, so as to subvert those relations which render it habi- 
table to man, or to give rise to great changes, at least, in the 
physical comfort of his state. To this the researches of 
geometers have enabled us to answer in the negative. A 
relation has been demonstrated by Lagrange between the 
masses, axes of the orbits, and excentricities of each planet, 
similar to what we have already stated with respect to their 
inclinations, viz. that if the mass of each planet be multiplied 
by the square root of the axis of its orbit, and the product by the 
square of its excentricity, the sum of all such products throughout 
the system is invariable; and as, in point of fact, this sum is ex- 
tremely small, so it will always remain. Now, since the axes 
of the orbits are liable to no secular changes, this is equiva- 
lent to saying that no one orbit shall increase its excentricity, 
unless at the expense of a common fund, the whole amount 
of which is, and must forever remain, extremely minute. 10 

10 There is nothing in this relation, however, taken per se, to secure the 
smaller planets — Mercury, Mars, Juno, Ceres, etc. — from a catastrophe, could 
they accumulate on themselves, or any one of them, the whole amount of this 
excentricity fund. But that can never be: Jupiter and Saturn will always re- 
tain the lion's share of it. A similar remark applies to the inclination fund of 
art. 639. These funds, be it observed, can never get into debt. Every term 
of them is essentially positive. 



602 OUTLINES OF ASTRONOMY 

(701 a.) (1865.) The actual numerical computation o! the 
limiting excentricities of the planets, taking into account all 
their mutual reactions, was attempted by Lagrange in 1782; 
but owing to an erroneous assumption of the mass of Venus, 
his results were rendered uncertain. M. Leverrier, in a re- 
markable memoir published in 1843, has resumed the sub- 
ject with the advantage of perfectly reliable data, and has 
obtained the following, as the superior limits of excentrici- 
ties of the seven principal then known planets — viz. for that 
of Mercury, 0-225646; Venus, 0-086716; the Earth, 0-077747; 
Mars, 0-142243; Jupiter, 0-061548; Saturn, 0-084919; and 
Uranus, 0*064646. And it is remarkable that although the 
erroneous assumption in question has vitiated Lagrange's 
conclusions as to the secular progression in the magnitudes 
of the excentricities, the superior limits arrived at by him 
agree very nearly indeed with these. For the inferior limit 
of that of the Earth's orbit, M. Leverrier assigns 0-003314, 
being the nearest approach it will make to the circular form. 
This will be attained in 23980 years from the epoch A.D. 
1800, for which the calculations are instituted; i.e. in A.D. 
25780. The triple period of the excentricities of Jupiter, 
Saturn, and Uranus, taken as a group, is 900,000 years (un- 
certain to 4000±). The maxima and the minima of that of 
Saturn are separated by an interval of 34647 years (uncertain 
to 117±), and its next minimum will happen in A.D. 17914, 
at which epoch its value will be 0-0136. In the Appendix 
the reader will find the elements of the earth's orbit, calcu- 
lated for intervals of 10,000 years from 100,000 years before 
A.D. 1800 to 100,000 after that date by M. Leverrier, and 
the excentricities by Mr. Croll for 1,000,000 years before and 
after the same epoch. 



OUTLINES OF ASTRONOMY 603 



CHAPTER XIV 

Of the Inequalities Independent of the Excentricities — The Moon's Varia- 
tion and Parallactic Inequality — Analogous Planetary Inequalities — 
Three Cases of Planetary Perturbation Distinguished — Of Inequalities 
Dependent on the Excentricities — Long Inequality of Jupiter and 
Saturn — Law of Reciprocity Between the Periodical Variations of the 
Elements of both Planets — Long Inequality of the Earth and Venus — 
Variation of the Epoch — Inequalities Incident on the Epoch Affecting 
the Mean Motion — Interpretation of the Constant Part of these In- 
equalities — Annual Equation of the Moon — Her Secular Acceleration 
— Lunar Inequalities Due to the Action of Venus — Effect of the Sphe- 
roid :tl Figure of the Earth and Other Planets on the Motions of their 
Satellites — Of the Tides — Masses of Disturbing Bodies Deducible from 
the Perturbations they Produce — Mass of the Moon, and of Jupiter's 
Satellites, how Ascertained — Perturbations of Uranus Resulting in 
the Discovery of Neptune — Determination of the Absolute Mass and 
Density of the Earth 

(702.) To calculate the actual place of a planet or the 
moon, in longitude and latitude at any assigned time, it is 
not enough to know the changes produced by perturbation 
in the elements of its orbit, still less to know the secular 
changes so produced, which are only the outstanding or un- 
compensated portions of much greater changes induced in 
short periods of configuration. We must be enabled to esti- 
mate the actual effect on its longitude of those periodical ac- 
celerations and retardations in the rate of its mean angular 
motion, and on its latitude of those deviations above and 
below the mean plane of its orbit, which result from the con- 
tinued action of the perturbative forces, not as compensated 
in long periods, but as in the act of their generation and de- 
struction in short ones. In this chapter we purpose to give 

an account of some of the most prominent of the equations 

Astronomy — Vol. XX — 7 



604 OUTLINES OF ASTRONOMY 

or inequalities thence arising, several of which are of high 
historical interest, as having become known by observation 
previous to the discovery of their theoretical causes, and as 
having, by their successive explanations from the theory of 
gravitation, removed what were in some instances regarded 
as formidable objections against that theory, and afforded in 
all most satisfactory and triumphant verifications of it. 

(703.) We shall begin with those which compensate 
themselves in a synodic revolution of the disturbed and 
disturbing body, and which are independent of any per- 
manent excentricity of either orbit, going through their 
changes and effecting their compensations in orbits slightly 
elliptic, almost precisely as if they were circular. These 
inequalities result, in fact, from a circulation of the true 
upper focus of the disturbed ellipse about its mean place 
in a curve whose form and magnitude the principles laid 
down in the last chapter enable us to assign in any pro- 
posed case. If the disturbed orbit be circular, this mean 
place coincides with its centre: if elliptic, with the situa- 
tion of its upper focus, as determined from the principles 
laid down in the last chapter. 

(704.) To understand the nature of this circulation, we 
must consider the joint action of the two elements of the 
disturbing force. Suppose H to be the place of the upper 
focus, corresponding to any situation P of the disturbed 
body, and let P P' be an infinitesimal element of its orbit, 
described in an instant of time. Then supposing no dis- 
turbing force to act, P P' will be a portion of an ellipse, 
having H for its focus, equally whether the point P or P' 
be regarded. But now let the disturbing forces act during 
the instant of describing P P\ Then the focus H will shift 
its position to H', to find which point we must recollect, 



OUTLINES OF ASTRONOMY 605 

1st. What is demonstrated in art. 671, vis. that the effect 
of the normal force is to vary the position of the line P' H 
so as to make the angle HPH' equal to double the varia- 
tion of the angle of tangency due to me action of that force, 
without altering the distance P H: so that in virtue of the 
normal force alone, H would move to a point A, along 
the line E Q, drawn from H to a point Q, 90° in advance 
of P (because S H being exceedingly small, the angle PHQ 
may be taken as a right angle when P S Q is so), H ap- 
proaching Q if the normal force act outward, but receding 




from Q if inward. And similarly the effect of the tangen- 
tial force (art. 670) is to vary the position of H in the direc- 
tion HPorP H, according as the force retards or acceler- 
ates P's motion. To find H' then from H draw H P, H Q, 
to P and to a point of P's orbit 90° in advance of P. On 
H Q take H h, the motion of the focus due to the normal 
force, and on K P take H k, the motion due to the tangential 
force; complete the parallelogram H H', and its diagonal 
H H' will be the element of the true path of H in virtue 
of the joint action of both forces. 

(705.) The most conspicuous case in the planetary system 
to which the above reasoning is applicable, is that of the 



606 OUTLINES OF ASTRONOMY 

moon disturbed by the sun. The inequalit}^ thus arising 
is known by the name of the moon's variation, and was 
discovered so early as about the year 975 by the Arabian 
astronomer Aboul Wefa. 1 Its magnitude (or the extent of 
fluctuation to and fro in the moon's longitude which it 
produces) is considerable, being no less than 1° 4', and 
it is otherwise interesting as being the first inequality 
produced by perturbation, which Newton succeeded in 
explaining by the theory of gravity. A good general idea 
of its nature may be formed by considering the direct action 
of the disturbing forces on the moon, supposed to move in 
a circular orbit. In such an orbit undisturbed, the velocity 
would be uniform; but the tangential force acting to ac- 
celerate her motion through the quadrants preceding her 
conjunction and opposition, and to retard it through the 
alternate quadrants, it is evident that the velocity will have 
two maxima and two minima, the former at the syzygies, 
the latter at the quadratures. Hence at the syzygies the 
velocity will exceed that which corresponds to a circular 
orbit, and at quadratures will fall short of it. The true 
orbit will therefore be less curved or more flattened than 
a circle in syzygies, and more curved \i.e. protuberant be- 
yond a circle) in quadratures. This would be the case even 
were the normal force not to act. But the action of that 
force increases the effect in question, for at the syzygies, 
and as far as 64° 14' on either side of them, it acts outward, 
or in counteraction of the earth's attraction, and thereby 
prevents the orbit from being so much curved as it other- 
wise would be; while at quadratures, and for 25° 46' on 
either side of them, it acts inward, aiding the earth's at- 

1 Sedillot, Nouvelles Recherches pour servir a l'Histoire de l'Astronomie 
chez les Arabes. 



OUTLINES OF ASTRONOMY 607 

traction, and rendering that portion of the orbit more 
curved than it otherwise would be. Thus the joint action 
of both forces distorts the orbit from a circle into a flattened 
or elliptic form, having the longer axis in quadratures, and 
the shorter in syzygies; and in this orbit the moon moves 
faster than with her mean velocity at syzygy (i.e. where 
she is nearest the earth) and slower at quadratures where 
furthest. Her angular motion about the earth is therefore 
for both reasons greater in the former than in the latter 
situation. Hence at syzygy her true longitude seen from 
the earth will be in the act of gaining on her mean — in 
quadratures of losing, and at some intermediate points (not 
very remote from the octants) will neither be gaining nor 
losing. But at these points, having been gaining or losing 
through the whole previous 90°, the amount of gain or loss 
will have attained its maximum. Consequently at the oc- 
tants the true longitude will deviate most from the mean in 
excess and defect, and the inequality in question is there- 
fore nil at syzygies and quadratures, and attains its maxima 
in advance or retardation at the octants, which is agreeable 
to observation. 

(706.) Let us, however, now see what account can be 
rendered of this inequality by the simultaneous variations 
of the axis and excentricity as above explained. The tan- 
gential force, as will be recollected, is nil at syzygies and 
quadratures, and a maximum at the octants, accelerative in 
the quadrants B A and D B, and retarding in A D and B 
E. In the two former then the axis is in process of length- 
ening; in the two latter, shortening. On the other hand 
the normal force vanishes at (a, b, d, e) 64° 14' from the 
syzygies. It acts outward over e A a, b B c?, and inward 
over a T> b and d E e. In virtue of the tangential force, 



608 



OUTLINES OF ASTRONOMY 



then, the point H moves toward P when P is in A D, B E, 
and from it when in D B, E A, the motion being nil when 
at A, B, D, E, and most rapid when at the octant D, at 
which points, therefore (so far as this force is concerned), 
the focus H would have its mean situation. And in virtue 
of the normal focus, the motion of H in the direction H Q 
will be at its maximum of rapidity toward Q at A, or B, 
from Q at D or E, and nil at a, b, d, e. It will assist us in 
following out these indications to obtain a notion of the 




form of the curve really described by H, if we trace sepa- 
rately the paths which H would pursue in virtue of either 
motion separately, since its true motion will necessarily 
result from the superposition of these partial motions, be- 
cause at every instant they are at right angles to each other, 
and therefore cannot interfere. First, then, it is evident, 
from what we have said of the tangential force, that when 
P is at A, H is for an instant at rest, but that as P removes 
from A toward D, H continually approaches P along their 
line of junction H P, which is, therefore, at each instant 



OUTLINES OF ASTRONOMY 

a tangent to the path of H. When P is in the octant, H 
is at its mean distance from P (equal to P S), and is then 
in the act of approaching P most rapidly. From thence to 
the quadrature D the movement of H toward P decreases 
in rapidity till the quadrature is attained, when H rests for 
an instant, and then begins to reverse its motiou, and travel 
from P at the same rate of progress as before toward it. 
Thus it is clear that, in virtue of the tangential force alone, 
H would describe a four-cusped curve, a, d, 6, e, its direc- 
tion of motion round S in this curve being opposite to that 
of P, so that A and a, D and d ) B and b ) E and e, shall be 
corresponding points. 

(707.) Next as regards the normal force. When the 
moon is at A. the motion of H is toward D, and is at its 




maximum of rapidity, but slackens as P proceeds toward D 
and as Q proceeds toward B. To the curve described, H Q 
will be always a tangent, and since at the neutral point of 
the normal force (or when P is 64° 14' from A, and Q 64° 
14' from D), the motion of H is for an instant nil and is then 



610 



OUTLINES OF ASTEOJSOlkl 



reversed, the curve will have a cusp at I corresponding, and 
H will then begin to travel along the arc I ra, while P de- 
scribes the corresponding arc from neutral point to neutral 
point through D. Arrived at the neutral point between 
D and B, the motion of H along Q H will be again arrested 
and reversed, giving rise to another cusp at m, and so on. 
Thus, in virtue of the normal force acting alone, the path 
of H would be the four-cusped, elongated curve I m n o, 
described with a motion round S the reverse of P's, and 
having a, d, b, e, for points corresponding to A, B, D, B, 
places of P. 

(708.) Nothing is now easier than to superpose these 
motions. Supposing Hj, H 2 to be the points in either curve 
corresponding to P, we nave nothing to do but to set from 
off S, S h equal and parallel to S R x in the one curve and 
from h, h H equal and parallel to S H 2 in the other. Let 
this be done for every corresponding point in the two 



I 



5 — <sl ^ «i^-~Tg 







curves, and there results an oval curve a db e, having for 

its semiaxis Sa = Sa x -f- S« 2 ; and &d = Sc? t -f- Sd 2 . And this 
will be the true path of the upper focus, the points a, d, b, e, 
corresponding to A, D, B, E, places of P. And from this it 
follows, 1st, that at A, B, the syzygies, the moon is in peri- 
gee in her momentary ellipse, the lower focus being nearer 
than the upper. 2dly, That in quadratures D, B, the moon 
is in apogee in her then momentary ellipse, the upper focus 



OUTLINES OF ASTRONOMY 611 

being then nearer than the lower. 3dly, That H revolves in 
the oval a db e the contrary way to P in its orbit, making a 
complete revolution from syzygy to syzygy in one synodic 
revolution of the moon. 

(709.) Taking 1 for the moon's mean distance from the 
earth, suppose we represent Sa^ or Sc?i (for they are equal) 
by 2a, Sa 2 by 26, and Sd. 2 by 2c, then will the semiaxes of 
the oval a d b e, Sa and Sc? be respectively la -\- 2b and 
2a -f- 2c, so that the excentricities of the momentary ellipses 
at A and D will be respectively a -\-b and a -f- c. The total 
amount of the effect of the tangential force on the axis, in 
passing from syzygy to quadrature, will evidently be equal 
to the length of the curvilinear arc a x d x {fig. art. 708), which 
is necessarily less than Sa x + Sc? x or 4a. Therefore the total 
effect on the semiaxis or distance of the moon is less than 
2a, and the excess and defect of the greatest and least values 
of this distance thus varied above and below the mean value 
SA = 1 (which call a) will be less than a. The moon then 
is moving at A in the perigee of an ellipse whose semiaxis is 
1 -f- a and excentricity a + 5, so that its actual distance from 
the earth there is 1 -f a — a — 6, which (because a is less 
than a) is less than 1 — b. Again, at D it is moving in apo- 
gee of an ellipse whose semiaxis is 1 — a and excentricity 
a -f c, so that its distance then from the earth is 1 — a -j- 
a -f- c, which (a being greater than a) is greater than 1 -f- c, 
the latter distance exceeding the former by 2a — 2a + b -f- c. 

(710.) Let us next consider the corresponding changes 
induced upon the angular velocity. Now it is a law of 
elliptic motion that at different points of different ellipses, 
each differing very little from a circle, the angular velocities 
are to each other as the square roots of the semiaxes directly, 
and as the squares of the distances inversely. In this case 



612 OUTLINES OF ASTRONOMY 

the semiaxes at A and D are to each other as 1 -f» a to 1 — a, 
or as 1 : 1 — 2a, so that their square roots are to each other 
as 1:1 — a. Again, the distances being to each other as 
1 -j- « __ a — b : 1 — a -{- a -\- c, the inverse ratio of their 
squares (since a, a, 6, c, are all very small quantities) is that 
of 1 — 2a + 2a + 2c : 1 + 2a — 2a — 26, or as 1 : 1 — 4a — 4a 
— 2b — ■ 2c. The angular velocities then are to each other in 
a ratio compounded of these two proportions, that is, in the 
ratio of 

1 : i + 3a — 4a — 2b — 2c, 

which is evidently that of a greater to a less quantity. It is 
obvious also, from the constitution of the second term of 
this ratio, that the normal force is far more influential in 
producing this result than the tangential. 

(711.) In the foregoing reasoning the sun has been re- 
garded as fixed. Let us now suppose it in motion (in a cir- 
cular orbit), then it is evident that at equal angles of elonga- 
tion (of P from M seen from S), equal disturbing forces, both 
tangential and normal, will act: only the syzygies and quad- 
ratures, as well as the neutral points of the normal force, 
instead of being points fixed in longitude on the orbit of the 
moon, will advance on that orbit with a uniform angular 
motion equal to the angular motion of the sun. The cuspi- 
dated curves a x d x b x e, and a 2 d a b 2 e 2 , Jig. art. 708, will, there- 
fore, no longer be re-entering curves; but each will have its 
cusps screwed round as it were in the direction of the sun's 
motion, so as to increase the angles between them in the 
ratio of the synodical to the sidereal revolution of the moon 
(art. 418). And if, in like manner, the motions in these two 
curves, thus separately described by H, be compounded, the 
resulting curve, though still (loosely speaking) a species of 
oval, will not return into itself, but will make successive 



OUTLINES OF ASTRONOMY 613 

spiroidal convolutions about S, its furthest and nearest 
points being in the same ratio more than 90° asunder. And 
to this movement that of the moon herself will conform, 
describing a species of elliptic spiroid, having its least dis- 
tances always in the line of syzygies and its greatest in that 
of quadratures. It is evident also, that, owing to the longer 
continued action of both forces, i.e. owing to the greater arc 
over which their intensities increase and decrease by equal 
steps, the branches of each curve between the cusps will be 
longer, and the cusps themselves will be more remote from 
S, and in the same degree will the dimensions of the result- 
ing oval be enlarged, and with them the amount of the in- 
equality in the moon's motion which they represent. 

(712.) In the above reasoning the sun's distance is sup- 
posed so great, that the disturbing forces in the semi- orbit 
nearer to it shall not sensibly differ from those in the more 
remote. The sun, however, is actually nearer to the moon 
in conjunction than in opposition by about one two-hun- 
dredth part of its whole distance, and this suffices to give 
rise to a very sensible inequality (called the parallactic in- 
equality) in the lunar motions, amounting to about 2' in its 
effect on the moon's longitude, and having for its period one 
synodical revolution or one lunation. As this inequality, 
though subordinate in the case of the moon to the great in- 
equality of the variation with which it stands in connection, 
becomes a prominent feature in the system of inequalities 
corresponding to it in the planetary perturbations (by reason 
of the very great variations of their distances from conjunc- 
tion to opposition), it will be necessary to indicate what 
modifications this consideration will introduce into the forms 
of our focus curves, and of their superposed oval. Eecur- 
ring then to our figures in arts. 706, 707, and supposing the 



61-1 OUTLINES OF ASTRONOMY 

moon to set out from E, and the upper focus, in each curve 
from e, it is evident that the intercuspidal arcs e a, a d, in 
the one, and e o, o a I, I d, in the other, being described 
under the influence of more powerful forces, will be greater 
than the arcs d b, b e, and d m, m b n, n e corresponding in 
the other half revolution. The two extremities of these 
curves then, the initial and terminal places of e in each, will 
not meet, and the same conclusion will hold respecting those 
of the compound oval in which the focus really revolves, 
which will, therefore, be as in the annexed figure. Thus, 
at the end of a complete lunation, the focus will have shifted 
its place from e to /in a line parallel to the line of quadra- 
tures. The next revolution, and the next, the same thing 



would happen. Meanwhile, however, the sun has advanced 
in its orbit, and the line of quadratures has changed its situ- 
ation by an equal angular motion. In consequence, the 
next terminal situation (g) of the forces will not lie in the 
line ef prolonged, but in a line parallel to the new situation 
of the line of quadratures, and this process continuing, will 
evidently give rise to a movement of circulation of the 
point e, round a mean situation in an annual period; and 
this, it is evident, is equivalent to an annual circulation of 
the central point of the compound oval itself, in a small 
orbit about its mean position S. Thus we see that no per- 
manent and indefinite increase of excentricity can arise from 
this cause; which would be the case, however, but for the 
annual motion of the sun. 

(713.) Inequalities precisely similar in principle to the 



OUTLINES OF ASTRONOMY 615 

variation and parallactic inequality of the moon, though 
greatly modified by the different relations of the dimensions 
of the orbits, prevail in all cases where planet disturbs 
planet. To what extent this modification is carried will be 
evident, if we cast our eyes on the examples given in art. 
612, where it will be seen that the disturbing force in con- 
junction often exceeds that in opposition in a very high 
ratio (being in the case of Neptune disturbing Uranus more 
than ten times as great). The effect will be, that the orbit 
described by the centre of the compound oval about S, will 
be much greater relatively to the dimensions of that oval 
itself, than in the case of the moon. Bearing in mind the 
nature and import of this modification, we majr proceed to 
inquire, apart from it, into the number and distribution of 
the undulations in the contour of the oval itself arising from 
the alternations of direction plus and minus of the disturb- 
ing forces in the course of a synodic revolution. But first 
it should be mentioned that, in the case of an exterior dis- 
turbed by an interior planet, the disturbing body's angular 
motion exceeds that of the disturbed. Hence P, though 
advancing in its orbit, recedes relatively to the line of syzy- 
gies, or, which comes to the same thing, the neutral points 
of either force overtake it in succession, and each, as it 
comes up to it, gives rise to a cusp in the corresponding 
focus curve. The angles between the successive cusps will 
therefore be to the angles between the corresponding neutral 
points for a fixed position of M, in the same constant ratio of 
the synodic to the sidereal period of P, which, however, is 
now a ratio of less inequality. These angles then will be 
contracted in amplitude, and, for the same reason as before, 
the excursions of the focus will be diminished, and the more 
so the shorter the synodic revolution. 



616 OUTLINES OF ASTRONOMY 

(714.) Since the cusps of either curve recur, in successive 
synodic revolutions in the same order, and at the same an- 
gular distances from each other, and from the line of con- 
junction, the same will be true of all the corresponding 
points in the curve resulting from their superposition. In 
that curve, every cusp, of either constituent, will give rise 
to a convexity, and every intercuspidal arc to a relative con- 
cavity. It is evident then that the compound curve or true 
path of the focus so resulting, but for the cause above men- 
tioned, would return into itself, whenever the periodic times 
of the disturbing and disturbed bodies are commensurate, 





because in that case the synodic period will also be com- 
mensurate with either, and the arc of longitude intercepted 
between the sidereal place of any one conjunction, and that 
next following it, will be an aliquot part of 360°. In all 
other cases it would be a non-re-entering, more or less undu- 
lating and more or less regular, spiroid, according to the 
number of cusps in each of the constituent curves (that is to 
say, according to the number of neutral points or changes of 
direction from inward to outward, or from accelerating to 
retarding, and vice versa, of the normal and tangential 
forces), in a complete synodic revolution, and their distri- 
bution over the circumference. 

(715.) With regard to these changes, it is necessary to 



OUTLINES OF ASTRONOMY 617 

distinguish three cases, in which the perturbations of planet 
by planet are very distinct in character. 1st. When the 
disturbing planet is exterior. In this case there are four 
neutral points of either force. Those of the tangential 
force occur at the syzygies, and at the points of the dis- 
turbed orbit (which we shall call points of equidistance), 
equidistant from the sun and the disturbing planet (at 
which points, as we have shown, art. 614, the total dis- 
turbing force is always directed inward toward the sun). 
Those of the normal force occur at points intermediate be- 
tween these last-mentioned points, and the syzygies, which, 
if the disturbing planet be very distant, hold nearly the sit- 
uation they do in the lunar theory, i.e. considerably nearer 
the quadratures than the syzygies. In proportion as the 
distance of the disturbing planet diminishes, two of these 
points, viz. those nearest the syzygy, approach to each 
other, and to the syzygy, and in the extreme case, when 
the dimensions of the orbits are equal, coincide with it. 
(716.) The second case is that in which the disturbing 
planet is interior to the disturbed, but at a distance from 
the sun greater than half that of the latter. In this case 
there are four neutral points of the tangential force, and 
only two of the normal. Those of the tangential force 
occur at the syzygies, and at the points of equidistance. 
The force retards the disturbed body from conjunction to 
the first such points after conjunction, accelerates it thence 
to the opposition, thence again retards it to the next point 
of equidistance, and finally again accelerates it up to the 
conjunction. As the disturbing orbit contracts in dimen- 
sion the points of equidistance approach; their distance 
from syzygy from 60° (the extreme case) diminishing to 
nothing, when they coincide with each other, and with the 



618 



OUTLINES OF ASTRONOMY 



conjunction. In the case of Saturn disturbed by Jupiter, 
that distance is only 23° 33'. The neutral points of the 
normal force lie somewhat beyond the quadratures, on the 
side of the opposition, and do not undergo any very 
material change of situation with the contraction of the 
disturbing orbit. 

(717.) The third case is that in which the diameter of 
the disturbing interior orbit is less than half that of the 
disturbed. In this case there are only two points of evan- 
escence for either force. Those of the tangential force are 
the syzygies. The disturbed planet is accelerated through- 
out the whole semi-revolution from conjunction to opposi- 




tion, and retarded from opposition to conjunction, the max- 
ima of acceleration and retardation occurring not far from 
quadrature. The neutral points of the normal force are 
situated nearly as in the last case; that is to say, beyond 
the quadratures toward the opposition. All these varieties 
the student will easily trace out by simply drawing the fig- 
ures, and resolving the forces in a series of cases, beginning 
with a very large and ending with a very small diameter of 
the disturbing orbit. It will greatly aid him in impressing 
on his imagination the general relations of the subject, if 



OUTLINES OF ASTRONOMY 619 

he construct, as he proceeds, for each case, the elegant and 
symmetrical ovals in which the points N and L (fig. art. 
675) always lie, for a fixed position of M, and of which 
the annexed figure expresses the forms they respectively 
assume in the third case now under consideration. The 
second only differs from this, in having the common vertex 
m of both ovals outside of the disturbed orbit A P, while 
in the case of an exterior disturbing planet the oval m L 
assumes a four-lobed form; its lobes respectively touching 
the oval m N in its vertices, and cutting the orbit A P 
in the points of equidistance aud of tangency (i.e. where 
M P S is a right angle) as in this figure. 




(718.) It would be easy now, bearing these features in 
mind, to trace in any proposed case the form of the spiroid 
carve, described, as above explained, by the upper focus. 
It will suffice, however, for our present purpose to remark, 
1st, That between every two successive conjunctions of P 
and M the same general form, the same subordinate undu- 
lations, and the same terminal displacement of the upper 
focus are continually repeated. 2dly, That the motion of 
the focus in this curve is retrograde whenever the disturb- 
ing planet is exterior, and that in consequence the apsides 
of the momentary ellipse also recede, with a mean velocity 
such as, but for that displacement, would bring them round 



620 OUTLINES OF ASTRONOMY 

at each conjunction to the same relative situation with re- 
spect to the line of syzygies. 3dly, That in consequence 
of this retrograde movement of the apse, the disturbed 
planet, apart from that consideration, would be twice in 
perihelio and twice in aphelio in its momentary ellipse 
in each synodic revolution, just as in the case of the 
moon disturbed by the sun — and that in consequence of 
this and of the undulating movement of the focus H itself, 
an inequality will arise, analogous, mutatis mutandis in each 
case, to the moon's variation, under which term we compre- 
hend (not exactly in conformity to its strict technical mean- 
ing in the lunar theory) not only the principal inequality 
thus arising, but all its subordinate fluctuations. And on 
this the parallactic inequality thus violently exaggerated is 
superposed. 

(719.) We come now to the class of inequalities which 
depend for their existence on an appreciable amount of 
permanent excentricity in the orbit of one or of both the 
disturbing and disturbed planets, in consequence of which 
all their conjunctions do not take place at equal distances 
either from the central body or from each other, and there- 
fore that symmetry in every synodic revolution on which 
depends the exact restoration of both the axis and excen- 
tricity to their original values at the completion of each 
such revolution no longer subsists. In passing from con- 
junction to conjunction, then, there will no longer be 
effected either a complete restoration of the upper focus 
to the same relative situation, or of the axis to the same 
length which they respectively had at the outset. At the 
same time it is not less evident that the differences in both 
respects are only what remain outstanding after the com- 
pensation of by far the greater part of the deviations to and 



OUTLINES OF ASTRONOMY 621 

fro from a mean state which occur in the course of the revo- 
lution; and that they amount to but small fractions of the 
total excursions of the focus from its first position, or of 
the increase and decrease in the length of the axis effected 
by the direct action of the tangential force — so small, in- 
deed, that, unless owing to peculiar adjustments they be 
enabled to accumulate again and again at successive con- 
junctions in the same direction, they would be altogether 
undeserving of any especial notice in a work of this nature. 
Such adjustments, however, would evidently exist if the 
periodic times of the planets were exactly commensurable; 
since in that case all the possible conjunctions which could 
ever happen (the elements not being materially changed) 
would take place at fixed points in longitude, the inter- 
mediate points being never visited by a conjunction. Now, 
of the conjunctions thus distributed, their relations to the 
lines of symmetry in the orbits being all dissimilar, some 
one must be more influential than the rest on each of the 
elements (not necessarily the same upon all). Consequently, 
in a complete cycle of conjunctions, wherein each has been 
visited in its turn, the influence of that one on the element 
to which it stands so especially related will preponderate 
over the counteracting and compensating influence of the 
rest, and thus, although in such a cycle as above specified, 
a further and much more exact compensation will have 
been effected in its value than in a single revolution; still 
that compensation will not be complete, but a portion of 
the effect (be it to increase or to diminish the excentricity 
or the axis, or to cause the apse to advance or to recede) 
will remain outstanding. In the next cycle of the same 
kind this will be repeated, and the result will be of the 
same character, and so on, till at length a sensible and 



622 OUTLINES OF ASTRONOMY 

ultimately a large amount of change shall have taken place, 
and in fact until the axis (and with it the mean motion) 
shall have so altered as to destroy the commensurability of 
periods, and the apsides have so shifted as to alter the place 
of the most influential conjunction. 

(720.) Now, although it is true that the mean motions 
of no two planets are exactly commensurate, yet cases are 
not wanting in which there exists an approach to this ad- 
justment. For instance, in the case of Jupiter and Saturn, 
a cycle composed of five periods of Jupiter and two of 
Saturn, although it does not exactly bring about the same 
configuration, does so pretty nearly. Five periods of Ju- 
piter are 21663 days, and two periods of Saturn, 21519 
days. The difference is only 146 days, in which Jupiter 
describes, on an average, 12°, and Saturn about 5°; so that 
after the lapse of the former interval they will only be 7° 
from a conjunction in the same parts of their orbits as 
before. If we calculate the time which will exactly bring 
about, on the average, three conjunctions of the two planets, 
we shall find it to be 21760 days, their synodical period 
being 7253*4 days. In this interval Saturn will have de- 
scribed 8° 6' in excess of two sidereal revolutions, and 
Jupiter the same angle in excess of five. Every third 
conjunction, then, will take place 8° 6' in advance of the 
preceding, which is near enough to establish, not, it is 
true, an identity with, but still a great approach to the 
case in question. The excess of action, for several such 
triple conjunctions (7 or 8) in succession, will lie the same 
way, and at each of them the elements of P's orbit and its 
angular motion will be similarly influenced, so as to accu- 
mulate the effect upon its longitude; thus giving rise to 
an irregularity of considerable magnitude and very long 



OUTLINES OF ASTRONOMY 623 

period, which is well known to astronomers by the name 
of the great inequality of Jupiter and Saturn. 

(721.) The arc 8° 6' is contained 44£ times in the whole 
circumference of 360°; and accordingly, if we trace round 
this particular conjunction, we shall find it will return to 
the same point of the orbit in so many times 21760 days, or 
in 2618 years. But the conjunction we are now consider- 
ing is only one out of three. The other two will happen at 
points of the orbit about 123° and 246° distant, and these 
points also will advance by the same arc of 8 C 6' in 21760 
days. Consequently the period of 2648 years will bring 
them all round, and in that interval each of them will pass 
through that point of the two orbits from which we com- 
menced: hence a conjunction (one or other of the three) will 
happen at that point once in one-third of this period, or in 
883 years; and this is, therefore, the cycle in which the 
"great inequality" would undergo its full compensation, 
did the elements of the orbits continue all that time in- 
variable. Their variation, however, is considerable in so 
long an interval; and, owing to this cause, the period 
itself is prolonged to about 918 years. 

(722.) We have selected this inequality as the most re- 
markable instance of this kind of action on account of its 
magnitude, the length of its period, and its high historical 
interest. It had long been remarked by astronomers, that 
on comparing together modern with ancient observations of 
Jupiter and Saturn, the mean motions of these planets did 
not appear to be uniform. The period of Saturn, for in- 
stance, appeared to have been lengthening throughout the 
whole of the seventeenth century, and that of Jupiter short- 
ening — that is to say, the one planet was constantly lagging 
behind, and the other getting in advance of its calculated 



0*21 OUTLINES OF ASTRONOMY 

place. On the other hand, in the eighteenth century, a 
process precisely the reverse seemed to be going on. It is 
true the whole retardations and accelerations observed were 
not very great; but, as their influence went on accumulat- 
ing, they produced, at length, material differences between 
the observed and calculated places of both these planets, 
which as they could not then be accounted for by any 
theory, excited a high degree of attention, and were even, 
at one time, too hastily regarded as almost subversive of the 
Newtonian doctrine of gravity. For a long while this differ- 
ence baffled every endeavor to account for it; till at length 
Laplace pointed out its cause in the near commensurability 
of the mean motions, as above shown, and succeeded in cal- 
culating its period and amount. 

(723.) The inequality in question amounts at its maxi- 
mum, to an alternate gain and loss of about 0° 49' in the 
longitude of Saturn, and a corresponding loss and gain of 
about 0° 21' in that of Jupiter. That an acceleration in the 
one planet must necessarily be accompanied by a retardation 
in the other, might appear at first sight self-evident, if we 
consider, that action and reaction being equal, and in con- 
trary directions, whatever momentum Jupiter communicates 
to Saturn in the direction P M, the same momentum must 
Saturn communicate to Jupiter in the direction M P. The 
one, therefore, it might seem to be plausibly argued, will be 
dragged forward, whenever the other is pulled back in its 
orbit. The inference is correct, so far as the general and 
final result goes; but the reasoning by which it would, on 
the first glance, appear to be thus summarily established is 
fallacious, or at least incomplete. It is perfectly true that 
whatever momentum Jupiter communicates directly to Sat- 
urn, Saturn communicates an equal momentum to Jupiter 



OUTLINES OF ASTRONOMY 625 

in an opposite linear direction. But it is not with the abso- 
lute motions of the two planets in space that we are now 
concerned, but with the relative motion of each separately, 
with respect to the sun regarded as at rest. The perturba- 
tive forces (the forces which disturb these relative motions) 
do not act along the line of junction of the planets (art. 614). 
In the reasoning thus objected to, the attraction of each on 
the sun has been left out of the account, 2 and it remains to 
be shown that these attractions neutralize and destroy each 
other's effects in considerable periods of time, as bearing 
upon the result in question. Suppose then that we for a 
moment abandon the point of view, in which we have 
hitherto all along considered the subject, and regard the 
sun as free to move, and liable to be displaced by the at- 
tractions of the two planets. Then will the movements of 
all be performed about the common centre of gravity, just 
as they would have been about the sun's centre regarded as 
immovable, the sun all the while circulating in a small orbit 
(with a motion compounded of the two elliptic motions it 
would have in virtue of their separate attractions) about the 
same centre. Now in this case M still disturbs P, and P, M, 
but the whole disturbing force now acts along their line of 
junction, and since it remains true that whatever momentum 
M generates in P, P will generate the same in M in a con- 
trary direction; it will also be strictly true that, so far as a 
disturbance of their elliptic motions about the common centre 
of gravity of the system is alone regarded, whatever disturb- 
ance of velocity is generated in the one, a contrary dis- 

2 We are here reading a sort of recantation. In the edition of 1833 the 
remarkable result in question is sought to be established by this vicious reason- 
ing. The mistake is a very natural one, and is so apt to haunt the ideas of be- 
ginners in this department of physics, that it is worth while expressly to warn 
them against it. 



626 OUTLINES OF ASTRONOMY 

turbance of velocity (only in the inverse ratio of the masses 
and modified, though never contradicted, by the directions 
in which they are respectively moving), will be generated in 
the other. Now when we are considering only inequalities 
of long period comprehending many complete revolutions 
of both planets, and which arise from changes in the axes of 
the orbits, affecting their mean motions, it matters not 
whether we suppose these motions performed about the 
common centre of gravity,- or about the sun, which never 
departs from that centre to any material extent (the mass of 
the sun being such in comparison with that of the planets, 
that that centre always lies within his surface). The mean 
motion therefore, regarded as the average angular velocity 
during a revolution, is the same whether estimated by refer- 
ence to the sun's centre, or to the centre of gravity, or, in 
other words, the relative mean motion referred to the sun is 
identical with the absolute mean motion referred to the cen- 
tre of gravity. 

(724.) This reasoning applies equally to every case of 
mutual disturbance resulting in a long inequality such as 
may arise from a slow and long-continued periodical increase 
and diminution of the axes, and geometers have accordingly 
demonstrated as a consequence from it, that the proportion 
in which such inequalities affect the longitudes of the two 
planets concerned, or the maxima of the excesses and de- 
fects of their longitudes above and below their elliptic 
values, thence arising, in each, are to each other in the in- 
verse ratio of their masses multiplied by the square roots 
of the major axes of their orbits, and this result is confirmed 
by observation, and will be found verified in the instance 
immediately in question as nearly as the uncertainty still 
subsisting as to the masses of the two planets will permit. 




OUTLINES OF ASTRONOMY 627 

(725.) The inequality in question, as has been observed 
in general (art. 718), would be much greater, were it not for 
the partial compensation which is operated m it in every 
triple conjunction of the planets. Suppose P Q B, to be 
Saturn's orbit, and/) q r Jupiter's; and suppose a conjunc- 
tion to take place at P/>, on the line S A; a second at 123° 
distance, on the line S B; a third at 246° distance, on S G; 
and the next at 368°, on S D. This last-mentioned con- 
junction, taking place nearly in the 
situation of the first, will produce 
nearly a repetition of the first effect 
in retarding or accelerating the plan- 
ets; but the other two, being in the 
most remote situations possible from 
the first, will happen under entirety 
different circumstances as to the posi- 
tion of the perihelia of the orbits. Now, we have seen that 
a presentation of the one planet to the other in conjunction, 
in a variety of situations, tends to produce compensation; 
and, in fact, the greatest possible amount of compensation 
which can be produced by only three conjunctions is when 
they are thus equally distributed round the centre. Hence 
we see that it is not the whole amount of perturbation which, 
is thus accumulated in each triple conjunction, out only that 
small part which is left uncompensated by the intermediate 
ones. The reader, who possesses already some acquaint- 
ance with the subject, will not be at a loss to perceive how 
this consideration is, in fact, equivalent to that part of the 
geometrical investigation of this inequality which leads us 
to seek its expression in terms of the third order, or involv- 
ing the cubes and products of three dimensions of the excen- 
tricities and inclinations: and how the continual accumula- 

ASTRONOMY — Vol. XX — 8 



628 OUTLINES OF ASTRONOMY 

tion of small quantities, during long periods, corresponds to 
what geometers intend when they speak of small terms re- 
ceiving great accessions of magnitude by the introduction of 
large coefficients in the process of integration. 

(726.) Similar considerations apply to every case of ap- 
proximate commensurability which can take place among 
the mean motions of any two planets. Such, for instance, 
is that which obtains between the mean motion of the earth 
and Yenus — 13 times the period of "Venus being very nearly 
equal to 8 times that of the earth. This gives rise to an ex- 
tremely near coincidence of every fifth conjunction, in the 
same parts of each orbit (within *^th part of a circumfer- 
ence), and therefore to a correspondingly extensive accumu- 
lation of the resulting uncompensated perturbation. But, 
on the other hand, the part of the perturbation thus accumu- 
lated is only that which remains outstanding after passing 
the equalizing ordeal of five conjunctions equally distrib- 
uted round the circle; or, in the language of geometers, is 
dependent on powers and products of the excentricities and 
inclinations of the fifth order. It is, therefore, extremely 
minute, and the whole resulting inequality, according to the 
elaborate calculations of Mr. Airy, to whom it owes its de- 
tection, amounts to no more than a few seconds at its maxi- 
mum, while its period is no less than 240 years. This 
example will serve to show to what minuteness these inqui- 
ries have been carried in the planetary theory. 

(727.) That variations of long period arising in the way 
above described are necessarily accompanied by similarly 
periodical displacements of the upper focus, equivalent in 
their effect to periodical fluctuations in the magnitude of the 
excentricity, and in the position of the line of apsides, is 
evident from what has been already said respecting the mo- 



OUTLINES OF ASTRONOMY 629 

tion of the upper focus under the influence of the disturbing 
forces. In the case of circular orbits the mean place of H 
coincides with S the centre of the sun, but if the orbits have 
any independent ellipticity, this coincidence will no longer 
exist — and the mean place of the upper focus will come to 
be inferred from the average of all the situations which it 
actually holds during an entire revolution Now the fixity 
of this point depends on the equality of each of the branches 
of the cuspidated curves, and consequent equality of excur- 
sion of the focus in each particular direction, in every suc- 
cessive situation of the line of conjunction. But if there be 




some one line of conjunction in which these excursions are 
greater in any one particular direction than in another, the 
mean place of the focus will be displaced, and if this process 
be repeated, that mean place will continue to deviate more 
and more from its original position, and thus will arise a 
circulation of the mean place of the focus for a revolution 
about another mean situation, the average of all the former 
mean places during a complete cycle of conjunctions. Sup- 
posing S to be the sun, the situation the upper focus 
would have, had these inequalities no existence, and H K 
the path of the upper focus, which it pursues about O by 



630 OUTLINES OF ASTRONOMY 

reason of them, then it is evident that in the course of a 
complete cycle of the inequality in question, the excentric- 
ity will have fluctuated between the extreme limits S J and 
S 1 and the direction of the longer axis between the extreme 
position S H and S K, and that if we suppose i j h k to be 
the corresponding mean places of the focus, ij will be the 
extent of the fluctuation of the mean excentricity, and the 
angle h s k, that of the longitude of the perigee. 

(728.) The periods then in which these fluctuations go 
through their phases are necessarily equal in duration with 
that of the inequality in longitude, with which they stand 
in connection. But it by no means follows that their max- 
ima all coincide. The variation of the axis to which that 
of the mean motion corresponds, depends on the tangential 
force only whose maximum is not at conjunction or opposi- 
tion, but at points remote from either, while the excentricity 
depends both on the normal and tangential forces, the maxi- 
mum of the former of which is at the conjunction. That 
particular conjunction, therefore, which is most influential 
on the axis, is not so on the excentricity, so that it can by 
no means be concluded that either the maximum value of 
the axis coincides with the maximum, or the minimum 
of the excentricity, or with the greatest excursion to or 
fro of the line of apsides from its mean situation, all that 
can be safely asserted is, that as either the axis or the 
excentricity of the one orbit varies, that of the other will 
vary in the opposite direction. 

(729.) The primary elements of the lunar and planetary 
orbits, which may be regarded as variable, are the longitude 
of the node, the inclination, the axis, excentricity, longi- 
tude of the perihelion, and epoch (art. 496). In the fore- 
going articles we have shown in what manner each of the 



OUTLINES OF ASTRONOMY 631 

first five of these elements is made to vary, by the direct 
action of the perturbing forces. It remains to explain in 
what manner the last comes to be affected by them. And 
here it is necessary, in the first instance, to remove some 
degree of obscurity which may be thought to hang about 
the sense in which the term itself is to be understood in 
speaking of an orbit, every other element of which is re- 
garded as in a continual state of variation. Supposing, 
then, that we were to reverse the process of calculation 
described in arts. 499 and 500 by which a planet's helio- 
centric longitude in an elliptic orbit is computed for a given 
time; and setting out with a heliocentric longitude ascer- 
tained by observation, all the other elements being known, 
we were to calculate either what mean longitude the planet 
had at a given epochal time, or, which would come to the 
same thing, at what moment of time (thenceforward to be 
assumed as an epoch) it had a given mean longitude. It 
is evident that by this means the epoch, if not otherwise 
known, would become known, whether we consider it as 
the moment of time corresponding to a convenient mean 
longitude, or as the mean longitude corresponding to a con- 
venient time. The latter way of considering it has some 
advantages in respect of general convenience, and astrono- 
mers are in agreement in employing, as an element under 
the title "Epoch of the mean longitude," the mean longi- 
tude of the planet so computed for a fixed date; as, for 
instance, the commencement of the year 1800, mean time 
at a given place. Supposing now all the elements of the 
orbit invariable, if we were to go through this reverse proc- 
ess, and thus ascertain the epoch (so defined) from any 
number of different perfectly correct heliocentric longi- 
tudes, it is clear we should always come to the same 



632 OUTLINES OF ASTRONOMY 

result. One and the same u epoch" would come out from 
all the calculations. 

(730.) Considering then the "epoch" in this light, as 
merely a result of this reversed process of calculation, and 
not as the direct result of an observation instituted for the 
purpose at the precise epochal moment of time (which would 
be, generally speaking, impracticable), it might be conceived 
subject to variation in two distinct ways, viz. dependently 
and independently. Dependently it must vary, as a neces- 
sary consequence of the variation of the other elements; 
because, if setting* out from one and the same observed 
heliocentric longitude of the planet, we calculate back to 
the epoch with two different sets of intermediate elements, 
the one set consisting of those which it had immediately 
before its arrival at that longitude, the other that which it 
takes up immediately after {i.e. with an unvaried and a 
varied system), we cannot (unless by singular accident of 
mutual counteraction) arrive at the same result; and the 
difference of the results is evidently the variation of the 
epoch. On the other hand, however, it cannot vary inde- 
pendently; for since this is the only mode in which the un- 
varied and varied epochs can become known, and as both re- 
sult from direct processes of calculation involving only given 
data, the results can only differ by reason of the difference 
of those data. Or we may argue thus. The change in the 
path of the planet, and its place in that path so changed, 
at any future time (supposing it to undergo no further 
variation), are entirely owing to the change in its velocity 
and direction, produced by the disturbing forces at the point 
of disturbance; now these latter changes (as we have above 
seen) are completely represented by the momentary change 
in the situation of the upper focus, taken in combination 



OUTLINES OF ASTRONOMY 633 

with the momentary variation in the plane of the orbit; and 
these therefore express the total effect of the disturbing 
forces. There is, therefore, no direct and specific action on 
the epoch as an independent variable. It is simply left 
to accommodate itself to the altered state of things in the 
mode already indicated. 

(731.) Nevertheless, should the effects of perturbation 
by inducing changes on these other elements affect the 
mean longitude of the planet in any other way than can 
be considered as properly taken account of, by the varied 
periodic time due to a change of axis, such effects must be 
regarded as incident on the epoch. This is the case with 
a very curious class of perturbations which we are now to 
consider, and which have their origin in an alteration of 
the average distance at which the disturbed body is found 
at every instant of a complete revolution, distinct from, 
and not brought about by the variation of the major semi- 
axis, or momentary "mean distance" which is an imaginary 
magnitude, to be carefully distinguished from the average 
of the actual distances now contemplated. Perturbations of 
this class (like the moon's variation, with which they are 
intimately connected) are independent on the excentricity 
of the disturbed orbit; for which reason we shall simplify 
our treatment of this part of the subject, by supposing that 
orbit to have no permanent excentricity, the upper focus in 
its successive displacements merely revolving about a mean 
position coincident with the lower. We shall also suppose 
M very distant, as in the lunar theory. 

(732.) Referring to what is said in arts. 706 and 707, and 
to the figures accompanying those articles, and considering 
first the effect of the tangential force, we see that besides 
the effect of that force in changing the length of the axis, 



634 OUTLINES OF ASTRONOMY 

and consequently the periodic time, it causes the upper 
focus H to describe, in each revolution of P, a four-cusped 
curve, a, d, b, e, about S, all whose intercuspidal arcs are 
similar and equal. This supposes M fixed, and at an in- 
variable distance — suppositions which simplify the relations 
of the subject, and (as we shall afterward show) do not 
affect the general nature of the conclusions to be drawn. 
In virtue, then, of the excentricity thus given rise to, P 
will be at the perigee of its momentary ellipse at syzygies 
and in its apogee at quadratures. Apart, therefore, from the 
change arising from the variation of axis, the distance of P 
from S will be less at syzygies, and greater at quadratures, 
than in the original circle. But the average of all the dis- 
tances during a whole revolution will be unaltered; because 
the distances of a, d, b, e from S being equal, and the arcs 
symmetrical, the approach in and about perigee will be 
equal to the recess in and about apogee. And, in like 
manner, the effect of the changes going on in the length 
of the axis itself, on the average in question, is nil, 
because the alternate increases and decreases of that 
length balance each other in a complete revolution. Thus 
we see that the tangential force is excluded from all in- 
fluence in producing the class of perturbations now under 
consideration. 

(733.) It is otherwise with respect to the normal force. 
In virtue of the action of that force the upper focus de- 
scribes, in each revolution of P, the four-cusped curve 
(fig. art. 707), whose intercuspidal arcs are alternately of 
very unequal extent, arising, as we have seen, from the 
longer duration and greater energy of the outward than 
of the inward action of the disturbing force. Although, 
therefore, in perigee at syzygies and in apogee at quadra- 



OUTLINES OF ASTRONOMY 635 

tures, the apogeal recess is much greater than the perigeal 
approach, inasmuch as S d greatly exceeds S a. On the 
average of a whole revolution, then, the recesses will pre- 
ponderate, and the average distance will therefore be 
greater in the disturbed than in the undisturbed orbit. 
And it is manifest that this conclusion is quite indepen- 
dent of any change in the length of the axis, which the 
normal force has no power to produce. 

(734.) But neither does the normal force operate any 
change of linear velocity in the disturbed body. When 
carried out, therefore, by the effect of that force to a 
greater distance from S, the angular velocity of its mo- 
tion round S will be diminished: and contrariwise when 
brought nearer. The average of all the momentary an- 
gular motions, therefore, will decrease with the increase 
in that of the momentary distances; and in a higher 
ratio, since the angular velocity, under an equable de- 
scription of areas, is inversely as the square of the 
distance, and the disturbing force, being (in the case 
supposed) directed to or from the centre, does not dis- 
turb that equable description (art. 616). Consequently, on 
the average of a whole revolution, the angular motion is 
slower, and therefore the time of completing a revolution, 
and returning to the same longitude, longer than in the 
undisturbed orbit, and that independent of and without any 
reference to the length of the momentary axis, and the 
"periodic time" or "mean motion" dependent thereon. 
We leave to the reader to follow out (as is easy to do) the 
same train of reasoning in the cases of planetary perturba- 
tion, when M is not very remote, and when it is interior to 
the disturbed orbit. In the latter case the preponderant 
effect changes from a retardation of angular velocity to an 



636 OUTLINES OF ASTRONOMY 

acceleration, and the dilatation of the average dimensions of 
P's orbit to a contraction. 

(735.) The above is an accurate analysis, according to 
strict dynamical principles, of an effect which, speaking 
roughly, may be assimilated to an alteration of M's gravi- 
tation toward S by the mean preponderant amount of the 
outward and inward action of the normal forces constantly 
exerted — nearly as would be the case if the mass of the dis- 
turbing body were formed into a ring of uniform thickness, 
concentric with S and of such diameter as to exert an action 
on P everywhere equal to such mean preponderant force, 
and in the same direction as to inward or outward. For it 
is clear that the action of such a ring on P, will be the 
difference of its attractions on the two points P and S, of 
which the latter occupies its centre, the former is excentric. 
Now the attraction of a ring on its centre is manifestly 
equal in all directions, and therefore, estimated in any one 
direction, is zero. On the other hand, on a point P out of 
its centre, if within the ring, the resulting attraction will 
always be outward, toward the nearest point of the ring, or 
directly from the centre. 3 But if P lie without the ring, the 

3 As this is a proposition which the equilibrium of Saturn's ring renders not 
merely speculative or illustrative, it will be well to demonstrate it; which may 
be done very simply, and without the aid of any calculus. Conceive a spherical 
shell, and a point within it; every line passing through the point, and terminat- 
ing both ways in the shell, will, of course, be equally inclined to its surface at 
either end, being a chord of a spherical surface, and, therefore symmetrically 
related to all its parts. Now, conceive a small double cone, or pyramid, having 
its apex at the point, and formed by the conical motion of such a line round the 
point. Then will the two portions of the spherical shell, which form the bases 
of both the cones, or pyramids, be similar and equally inclined to their axes. 
Therefore their areas will be to each other as the squares of their distance from 
the common apex. Therefore their attractions on it will be equal, because the 
attraction is as the attracting matter directly, and the square of its distance in- 
versely. Now, these attractions act in opposite directions, and therefore coun- 
teract each other. Therefore the point is in equilibrium between them; and as 
the same is true of every such pair of areas into which the spherical shell can 
be broken up, therefore the point will be in equilibrium however situated within 
such a spherical shell. Now take a ring, and treat it similarly, breaking its 



OUTLINES OF ASTRONOMY 637 

resulting force will act always inward, urging P toward its 
centre. Hence it appears that the mean effect of the radial 
force of the ring will be different in its direction, according 
as the orbit of the disturbing body is exterior or interior 
to that of the disturbed. In the former case it will act in. 
diminution, in the latter in augmentation of the central 
gravity. 

(736.) Kegarding, still, only the mean effect, as produced 
in a great number of revolutions of both bodies, it is evi- 
dent that such an increase of central force will be accom- 
panied with a diminution of periodic time and distance of a 
body revolving with a stated velocity, and vice versa. This, 
as we have shown, is the first and most obvious effect of the 
radial part of the disturbing force, when exactly analyzed. 
It alters permanently, and by a certain mean amount, the 
distances and times of revolution of all the bodies compos- 
ing the planetary system, from what they would be, did 
each planet circulate about the sun uninfluenced by the at- 
traction of the rest; the angular motion of the interior bodies 
of the system being thus rendered less, and those of the exte- 
rior greater, than on that supposition. The latter effect, in- 
deed, might be at once concluded from this obvious consider- 
ation — that all the planets revolving interiorly to any orbit 
may be considered as adding to the general aggregate of the 
attracting matter within, which is not the less efficient for 
being distributed over space, and maintained in a state of 
circulation. 

circumference up into pairs of elements, the bases of triangles formed by lines 
passing through the attracted point. Here the attracting elements being lines, 
not surfaces, are in the simple ratio of the distances, not the duplicate, as they 
should be to maintain the equilibrium. Therefore it will not be maintained, 
but the nearest elements will have the superiority, and the point will, on the 
whole, be urged toward the nearest part of the ring. The same is true of every 
linear ring, and is therefore true of any assemblage of concentric ones forming 
ft flat annulus, like the ring of Saturn. 



688 OUTLINES OF ASTRONOMY 

(737.) This effect, however, is one which we have no 
means of measuring, or even of detecting, otherwise than by 
calculation. For our knowledge of the periods of the plan- 
ets is drawn from observations made on them in their actual 
state, and therefore under the influence of this, which may 
be regarded as a sort of constant part of the perturbative 
action. Their observed mean motions are therefore affected 
by the whole amount of its influence; and we have no 
means of distinguishing this by observation from the direct 
effect of the sun's attraction, with which it is blended. Our 
knowledge, however, of the masses of the planets assures us 
that it is extremely small ; and this, in fact, is ail which it 
is at all important to us to know, in the theory of their 
motions. 

(738.) The action of the sun upon the moon, in like man- 
ner, tends, by its mean influence during many successive 
revolutions of both bodies, to increase permanently the 
moon's distance and periodic time. But this general aver- 
age is not established, either in the case of the moon or 
planets, without a series of subordinate fluctuations, which 
we have purposely neglected to take account of in the above 
reasoning, and which obviously tend, in the average of a 
great multitude of revolutions, to neutralize each other. In 
the lunar theory, however, some of these subordinate fluc- 
tuations are very sensible to observation. The most con- 
spicuous of these is the moon's annual equation; so called 
because it consists in an alternate increase and decrease in 
her longitude, corresponding with the earth's situation in 
its annual orbit; i.e. to its angular distance from the perihe- 
lion, and therefore having a year instead of a month, or 
aliquot part of a month, for its period. To understand the 
mode of its production, let us suppose the sun, still holding 



OUTLINES OF ASTRONOMY 639 

a fixed position in longitude, to approach gradually nearer 
to the earth. Then will all its disturbing forces be gradu- 
ally increased in a very high ratio compared with the dimi- 
nution of the distance (being inversely as its cube; so that 
its effects of every kind are three times greater in respect of 
any change of distance, than they would be by the simple 
law of proportionality). Hence, it is obvious that the focus 
H (art. 707) in the act of describing each intercuspidal arc of 
the curve a, d, b, <?, will be continually carried out further 
and further from S; and the curve, instead of returning into 
itself at the end of each revolution, will open out into a sort 
of cuspidated spiral, as in the figure annexed. Ketracing 
now the reasoning of art. 733 as adapted 'to this state of 
things, it will be seen that so long as 
this dilatation goes on, so long will the 
differences between M's recess from S in 
aphelio and its approach in perihelio 
(which is equal to the difference of 
consecutive long and short semidiameters of this curve) 
also continue to increase, and with it the average of the 
distances of M from S in a whole revolution, and con- 
sequently also the time of performing such a revolution. 
The reverse process will go on as the sun again recedes. 
Thus it appears that, as the sun approaches the earth, the 
mean angular motion of the moon on the average of a whole 
revolution will diminish, and the duration of each lunation 
will therefore exceed that of the foregoing, and vice versa. 
(739.) The moon's orbit being supposed circular, the 
sun's orbitual motion will have no other effect than to keep 
the moon longer under the influence of every gradation of 
the disturbing force, than would have been the case had his 
situation in longitude remained unaltered (art. 711). The 




640 OUTLINES OF ASTRONOMY 

same effects, therefore, will take place, only on an increased 
scale in the proportion of the increased time; i.e. in the pro- 
portion of the synodic to the sidereal revolution of the 
moon. Observation confirms these results, and assigns to 
the inequality in question a maximum value of between 10' 
and 11', by which the moon is at one time in advance of, 
and at another behind, its mean place, in consequence of 
this perturbation. 

(740.) To this class of inequalities we must refer one of 
great importance, and extending over an immense period of 
time, known by the name of the secular acceleration of the 
moon's mean motion. It had been observed by Dr. Halley, 
on comparing together the records of the most ancient lunar 
eclipses of the Chaldean astronomers with those of modern 
times, that the period of the moon's revolution at present is 
sensibly shorter than at that remote epoch; and this result 
was confirmed by a further comparison of both sets of obser- 
vations with those of the Arabian astronomers of the eighth 
and ninth centuries. It apj)eared, from these comparisons, 
that the rate at which the moon's mean motion increases is 
about 11 seconds per century — a quantity small in itself, 
but becoming considerable by its accumulation during a 
succession of ages. This remarkable fact, like the great 
equation of Jupiter and Saturn, had been long the subject 
of toilsome investigation to geometers. Indeed, so difficult 
did it appear to render any exact account of, that while some 
were on the point of again declaring the theory of gravity 
inadequate to its explanation, others were for rejecting alto- 
gether the evidence on which it rested, although quite as 
satisfactory as that on which most historical events are 
credited. It was in this dilemma that Laplace once more 
stepped in to rescue physical astronomy from its reproach, 



OUTLINES OF ASTRONOMY 641 

by pointing out the real cause of the phenomenon in ques- 
tion, which, when so explained, is one of the most curious 
and instructive in the whole range of our subject — one 
which leads our speculations further into the past and 
future, and points to longer vistas in the dim perspective 
of changes which our system has undergone and is yet to 
undergo, than any other which observation assisted by 
theory has developed. 

(741.) The year is not an exact number of lunations. It 
consists of twelve and a fraction. Supposing then the sun 
and moon to set out from conjunction together; at the 
twelfth conjunction subsequent the sun will not have re- 
turned precisely to the same point of its annual orbit, but 
will fall somewhat short of it, and at the thirteenth will 
have overpassed it. Hence in twelve lunations the gain of 
longitude during the first half year will be somewhat under 
and in thirteen somewhat over compensated. In twenty-six 
it will be nearly twice as much overcompensated, in thirty- 
nine not quite so nearly three times as much, and so on, 
until, after a certain number of such multiples of a lunation 
have elapsed, the sun will be found half a revolution in ad- 
vance, and in place of receding further at the expiration of 
the next, it will have begun to approach. From this time 
every succeeding cycle will destroy some portion of that 
overcompensation, until a complete revolution of the sun in 
excess shall be accomplished. Thus arises a subordinate 
or rather supplementary inequality, having for its period as 
many years as is necessary to multiply the deficient arc into 
a whole revolution, at the end of which time a much more 
exact compensation will have been operated, and so on. 
Thus after a moderate number of years an almost perfect 
compensation will be effected, and if we extend our views 



642 OUTLINES OF ASTRONOMY 

to centuries we may consider it as quite so. Such at least 
would be the case if the solar ellipse were invariable. But 
that ellipse is kept in a continual but excessively slow state 
of change by the action of the planets on the earth. Its 
axis, it is true, remains unaltered; but its excentricity is, 
and has been since the earliest ages, diminishing; and this 
diminution will continue, as is explained in art. 701 a, till 
the excentricity has attained its minimum value, 0-003314; 
after which it will again open out into an ellipse, increasing 
in excentricity up to 0-077747, and then again decrease. 
The time required for these evolutions, though calculable, 
has not been calculated, further than to satisfy us that it is 
not to be reckoned by hundreds or by thousands of years. 
It is a period, in short, in which the whole history of as- 
tronomy and of the human race occupies but as it were a 
point, during which all its changes are to be regarded as 
uniform. rTow, it is by this variation in the excentricity of 
the earth's orbit that the secular acceleration of the moon is 
caused. The compensation above spoken of (even after the 
lapse of centuries) will now, we see, be only imperfectly 
effected, owing to this slow shifting of one of the essential 
data. The steps of restoration are no longer identical with, 
nor equal to, those of change. The struggle up hill is not 
maintained on equal terms with the downward tendency. 
The ground is all the while slowly sliding beneath the feet 
of the antagonists. During the whole time that the earth's 
excentricity is diminishing, a preponderance is given to the 
reaction over the action; and it is not till that diminution 
shall cease, that the tables will be turned, and the process of 
ultimate restoration will commence. Meanwhile, a minute, 
outstanding, and uncompensated effect in favor of accelera- 
tion is left at each recurrence, or near recurrence, of the 



OUTLINES OF ASTRONOMY 643 

same configurations of the sun, the moon, and the solar peri- 
gee. These accumulate, and at length affect the moon's 
longitude to an extent not to be overlooked. 

(742.) The phenomenon, of which we have now given an 
account, is another and very striking example of the propa- 
gation of a periodic change from one part of a system to an- 
other. The planets, with one exception, have no direct ap- 
preciable action on the lunar motions as referred to the 
earth. Their masses are too small, and their distances too 
great, for their difference of action on the moon and earth 
ever to become sensible. Yet their effect on the earth's 
orbit is thus, we see, propagated through the sun to that of 
the moon; and, what is very remarkable, the transmitted 
effect thus indirectly produced on the angle described by 
the moon round the earth is more sensible to observation 
than that directly produced by them on the angle described 
by the earth round the sun. 

(743.) Referring to the reasoning of art. 738, we shall 
perceive that if, owing to any other cause than its elliptic 
motion, the sun's distance from the earth be subject to a 
periodical increase and decrease, that variation will give rise 
to a lunar inequality of equal period analogous to the an- 
nual equation. It thus happens that very minute changes 
impressed on the orbit of the earth, by the direct action of 
the planets (provided their periods, though not properly 
speaking secular, be of considerable length), may make 
themselves sensible in the lunar motions. The longitude 
of that satellite, as observed from the earth, is, in fact, sin- 
gularly sensible to this kind of reflected action, which illus- 
trates in a striking manner the principle of forced vibrations 
laid down in art. 650. The reason of this will be readily 
apprehended, if we consider that however trifling the in- 



644 OUTLINES OF ASTRONOMY 

crease of her longitude which would arise in a single revo- 
lution, from a minute and almost infinitesimal increase of 
her mean angular velocity, that increase is not only repeated 
in each subsequent revolution, but is reinforced during each 
by a similar fresh accession of angular motion generated in 
its lapse. This process goes on so long as the angular mo- 
tion continues to increase, and only begins to be reversed 
when lapse of time, bringing round a contrary action on the 
angular motion, shall have destroyed the excess of velocity 
previously gained, and begun to operate a retardation. In 
this respect, the advance gained by the moon on her undis- 
turbed place may be assimilated, during its increase, to the 
space described from rest under the action of a continually 
accelerating force. The velocity gained in each instant is 
not only effective in carrying the body forward during each 
subsequent instant, but new velocities are every instant 
generated, and go on adding their cumulative effects to 
those before produced. 

(744.) The distance of the earth from the sun, like that 
of the moon from the earth, may be affected in its average 
value estimated over long periods embracing many revolu- 
tions, in two modes, conformably to the theory above deliv- 
ered. 1st, it may vary by a variation in the length of the 
axis major of its orbit, arising from the direct action of some 
tangential disturbing force on its velocity, and thereby pro- 
ducing a change of mean motion and periodic time in virtue 
of the Keplerian law of periods, which declares that the 
periodic times are in the sesquiplicate ratio of the mean dis- 
tances. Or, 2dly, it may vary by reason of that peculiar 
action on the average of actual distances during a revolu- 
tion, which arises from variations of excentricity and peri- 
helion only, and which produces that sort of change in the 



OUTLINES OF ASTRONOMY 645 

mean motion which we have characterized as incident on 
the epoch. The change of mean motion thus arising, h.as 
nothing whatever to do with any variation of the major axis. 
It does not depend on the change of distance by the Kep- 
lerian law of periods, but by that of areas. The altered 
mean motion is not sub-sesquiplicate to the altered axis of 
the ellipse, which in fact does not alter at all, but is sub- 
duplicate to the altered average of distances in a revolution; a 
distinction which must be carefully borne in mind by every 
one who will clearly understand either the subject itself, or 
the force of Newton's explanation of it in the 6th Corollary 
of his celebrated 66th Proposition. In whichever mode, 
however, an alteration in the mean motion is effected, if we 
accommodate the general sense of onr language to the spe- 
cialties of the case, it remains true that every change in the 
mean motion is accompanied with a corresponding change 
in the mean distance. 

(745.) Now we have seen (art. 726) that Yenus produces 
in the earth a perturbation in longitude, of so long a period 
(240 years) that it cannot well be regarded without violence 
to ordinary language, otherwise than as an equation of the 
mean motion. Of course, therefore, it follows that during 
that half of this long period of time, in which the earth's 
motion is retarded, the distance between the sun and earth 
is on the increase, and vice versa. Minute as is the equation 
in question, and consequent alteration of solar distance, and 
almost inconceivably minute as is the effect produced on the 
moon's mean angular velocity in a single lunation, yet the 
great number of lunations (1484), during which the effect 
goes on accumulating in one direction, causes the moon at 
the moment when that accumulation has attained its maxi- 
mum to be very sensibly in advance of its undisturbed place 



646 OUTLINES OF ASTRONOMY 

(viz. by 23" of longitude), and after 1484: more lunations, as 
much in arrear. The calculations by which this curious 
result has been established, formidable from their length 
and intricacy, are due to the industry, as the discovery of 
its origin is to the sagacity, of Professor Hansen. 

(746.) The action of Yenus, just explained, is indirect, 
being as it were a sort of reflection of its influence on the 
earth's orbit. But a very remarkable instance of its influ- 
ence, in actually perturbing the moon's motions by its direct 
attraction, has been pointed out, and the inequality due to 
it computed by the same eminent geometer. 4 As the details 
of his processes have not yet appeared, we can here only ex- 
plain, in general terms, the principle on which the result in 
question depends, and the nature of the peculiar adjustment 
of the mean angular velocities of the earth and Venus which 
render it effective. The disturbing forces of Yenus on the 
moon are capable of being represented or expressed (as is 
indeed generally the case with all the forces concerned in 
producing planetary disturbance) by the substitution for 
them of a series of other forces, each having a period or 
cycle within which it attains a maximum in one direction, 
decreases to nothing, reverses its action, attains a maximum 
in the opposite direction, again decreases to nothing, again 
reverses its action and reattains its former magnitude, and 
so on. These cycles differ for each particular constituent 
or term, as it is called, of the total forces considered as so 
broken up into partial ones, and generally speaking, every 
combination which can be formed by subtracting a multiple 
of the mean motion of one of the bodies concerned from a 
multiple of that of the other, and, when there are three 

4 Astronomische Nachrichten, No. 597. 



OUTLINES OF ASTRONOMY 647 

bodies disturbing one another, every such triple combina- 
tion becomes, under the technical name of an argument, the 
cyclical representative of a force acting in the manner and 
according to the law described. Each of these periodically 
acting forces produces its perturbative effect, according to 
the law of the superposition of small motions, as if the 
others had no existence. And if it happen, as in an im- 
mense majority of cases it does, that the cycle of any par- 
ticular one of these partial forces has no relation to the 
periodic time of the disturbed body, so as to bring it to the 
same, or very nearly the same point of its orbit, or to any 
situation favorable to any particular form of disturbance, 
over and over again when the force is at its maximum; that 
force will, in a few revolutions, neutralize its own effect, 
and nothing but fluctuations of brief duration can result 
from its action. The contrary will evidently be the case, 
if the cycle of the force coincide so nearly with the cycle 
of the moon's anomalistic revolution, as to bring round the 
maximum of the force acting in one and the same direction 
(whether tangential or' normal) either accurately, or very 
nearly indeed to some definite point, as, for example, the 
apogee of her orbit. Whatever the effect produced by such 
a force on the angular motion of the moon, if it be not ex- 
actly compensated in one cycle of its action, it will go on 
accumulating, being repeated over and over again under 
circumstances very nearly the same, for many successive 
revolutions, until at length, owing to the want of precise 
accuracy in the adjustment of that cycle to the anomalistic 
period, the maximum of the force (in the same phase of its 
action) is brought to coincide with a point in the orbit (as 
the perigee), determinative of an opposite effect, and thus, 
at length, a compensation will be worked out; in a time, 



64:8 OUTLINES OF ASTRONOMY 

however, so much the longer as the difference between the 
cycle of the force and the moon's anomalistic period is less. 
(747.) Now, in fact, in the case of Venus disturbing the 
moon, there exists a cyclical combination of this kind. Of 
course the disturbing force of Venus on the moon varies 
with her distance from the earth, and this distance again 
depends on her configuration with respect to the earth and 
"the sun, taking into account the ellipticity of both their orbits. 
Among the combinations which take their rise from this 
latter consideration, and which, as may easily be s ap- 
posed, are of great complexity, th'ere is a term (an exceed- 
ingly minute one), whose argument or cycle is determined 
by subtracting 16 times the mean motion of the earth 
from 18 times that of Venus. The difference is so 
very nearly the mean motion of the moon in her anom- 
alistic revolution, that whereas the latter revolution is 
completed in 27 d 13 h 18 m 32 -3 s , the cycle of the force is 
completed in 27 d 13 h 7 m 35 '6 s , differing from the other by 
no more than 10 m 56 -7 s , or about one 3625th part of a com- 
plete period of the moon from apogee to apogee. During 
half of this very long interval (that is to say, during about 
136i years), the perturbations produced by a force of this 
character, go on increasing and accumulating, and are de- 
stroyed in another equal interval. Although therefore 
excessively minute in their actual effect on the angular 
motion, this minuteness is compensated by the number of 
repeated acts of accumulation, and by the length of time 
during which they continue to act on the longitude. Ac- 
cordingly M. Hansen has found the total amount- of fluctua- 
tion to and fro, or the value of the equation of the moon's 
longitude, so arising to be 274*. It is exceedingly interest- 
ing to observe that the two equations considered in these. 



OUTLINES OF ASTRONOMY 

latter paragraphs, account satisfactorily for the only remain- 
ing material differences between theory and observation in 
the modern history of this hitherto rebellious satellite. We 
have not thought it necessary (indeed it would have re- 
quired a treatise on the subject) to go into a special account 
of the almost innumerable other lunar inequalities which 
have been computed and tabulated, and which are neces- 
sary to be taken into account in every computation of her 
place from the tables. Many of them are of very much 
larger amount than these. We ought not, however, to pass 
uimoticed, that the parallactic inequality, already explained 
(art. 712), is interesting, as affording a measure of the sun's 
distance. For this equation originates, as there shown, in 
the fact that the disturbing forces are not precisely alike 
in the two halves of the moon's orbit nearest to and most 
remote from the sun, all their values being greater in the 
former half. As a knowledge of the relative dimensions of 
the solar and lunar orbits enables us to calculate d priori 
the amount of this inequality, so a knowledge of that amount 
deduced by the comparison of a great number of observed 
places of the moon with tables in which every inequality 
but this should be included, would enable us conversely to 
ascertain the ratio of the distances in question. Owing 
to the smallness of the inequality, this is not a very accu- 
rate mode of obtaining an element of so much importance 
in astronomy as the sun's distance, but were it larger (i.e. 
were the moon's orbit considerably larger than it actually 
is), this would be, perhaps, the most exact method of any 
by which it could be concluded. 

(748.) The greatest of all the lunar inequalities, pro- 
duced by perturbation, is that called the evection. It arises 
directly from the variation of the excentricity of her orbit, 



650 OUTLINES OF ASTRONOMY 

and from the fluctuation to and fro in the general progress 
of the line of apsides, caused by the different situation of 
the sun, with respect to that line (arts. 685, 691). Owing 
to these causes the moon is alternately in advance, and in 
arrear of her elliptic place by about 1° 20' 30". This equa- 
tion was known to the ancients, having been discovered by 
Ptolemy, by the comparison of a long series of observations 
handed down to him from the earliest ages of astronomy. 
The mode in which the effects of these several sources of 
inequality become grouped together under one principal 
argument common to them all, belongs, for its explanation, N 
rather to works specially treating of the lunar theory than 
to a treatise of this kind. 

(749.) Some small perturbations are produced in the lunar 
orbit by the protuberant matter of the earth's equator. The 
attraction of a sphere is the same as if all its matter were 
condensed into a point in its centre; but that is not the case 
with a spheroid. The attraction of such a mass is neither 
exactly directed to its centre, nor does it exactly follow the 
law of the inverse squares of the distances. Hence will 
arise a series of perturbations, extremely small in amount, 
but still perceptible in the lunar motions; by which the 
node and the apogee will be affected. A more remarkable 
consequence of this cause, however, is a small nutation of 
the lunar orbit, exactly analogous to that which the moon 
causes in the plane of the earth's equator, by its action on 
the same elliptic protuberance. And, in general, it may be 
observed, that in the systems of planets which have satel- 
lites, the elliptic figure of the primary has a tendency to 
bring the orbits of the satellites to coincide with its equator 
— a tendency which, though small in the case of the earth, 
yet in that of Jupiter, whose ellipticity is very considerable, 



OUTLINES OF ASTRONOMY 651 

and of Saturn especially, where the elliptic! ty of the body 
is reinforced by the attraction of the rings, becomes pre- 
dominant over every external and internal cause of disturb- 
ance, and produces and maintains an almost exact coinci- 
dence of the planes in question. Such, at least, is the case 
with the nearer satellites. The more distant are compara- 
tively less affected by this cause, the difference of attrac- 
tions between a sphere and spheroid diminishing with great 
rapidity as the distance increases. Thus, while the orbits 
of all the interior satellites of Saturn lie almost exactly in 
the plane of the ring and equator of the planet, that of the 
external satellite, whose distance from Saturn is between 
sixty and seventy diameters of the planet, is inclined to that 
plane considerably. On the other hand, this considerable 
distance, while it permits the satellite to retain its actual 
inclination, prevents (by parity of reasoning) the ring and 
equator of the planet from being perceptibly disturbed by 
its attraction, or being subjected to any appreciable move- 
ments analogous to our nutation and precession. If such 
exist, they must be much slower than those of the earth; 
the mass of this satellite being, as far as can be judged by 
its apparent size, a much smaller fraction of that of Saturn 
than the moon is of the earth; while the solar precession, 
by reason of the immense distance of the sun, must be quite 
imperceptible. 

(750.) The subject of the tides, though rather belonging 
to terrestrial physics than properly to astronomy, is yet so 
directly connected with the theory of the lunar perturba- 
tions, that we cannot omit some explanatory notice of it, 
especially since many persons find a strange difficulty in 
conceiving the manner in which they are produced. That 

the sun, or moon, should by its attraction heap up the waters 

Astronomy — Vol. XX — 9 



652 OUTLINES OF ASTRONOMY 

of the ocean under it, seems to them very natural. That it 
should at the same time heap them up on the opposite side 
seems, on the contrary, palpably absurd. The error of this 
class of objectors is of the same kind with that noticed in 
art. 723, and consists in disregarding the attraction of the 
disturbing body on the mass of the earth, and looking on 
it as wholly effective on the superficial water. Were the 
earth indeed absolutely fixed, held in its place by an ex- 
ternal force, and the water left free to move, no doubt the 
effect of the disturbing power would be to produce a single 
accumulation vertically under the disturbing body. But it 
is not by its whole attraction, but by the difference of its 
attractions on the superficial water at both sides, and on the 
central mass, that the waters are raised: just as in the theory 
of the moon, the difference of the sun's attractions on the 
moon and on the earth (regarded as movable and as obey- 
ing that amount of attraction which is due to its situation) 
gives rise to a relative tendency in the moon to recede from 
the earth in conjunction and opposition, and to approach it 
in quadratures. Referring to the figure of art. 675, instead 
of supposing A D B E to represent the moon's orbit, let it 
be supposed to represent a section of the (comparatively) 
thin film of water reposing on the globe of the earth, in 
a great circle, the plane of which passes through the dis- 
turbing body M, which we shall suppose to be the moon. 
The disturbing force on a particle at P will then (exactly 
as in the lunar theory) be represented in amount and direc- 
tion by N S, on the same scale on which S M represents the 
moon's whole attraction on a particle situated at S. This 
force, applied at P, will urge it in the direction P X parallel 
toN S; and therefore, when compounded with the direct 
force of gravity which (neglecting as of no account in this 



OUTLINES OF ASTRONOMY 



653 



theory the spheroidal form of the earth) urges P toward S, 
will be equivalent to a single force deviating from the 
direction P S toward X. Suppose P T to be the direction 
of this force, which, it is easy to see, will be directed toward 
a point in B S produced, at an extremely small distance 
below S, because of the excessive minuteness of the dis- 
turbing force compared to gravity. 6 Then if this be done 
at every point of the quadrant A D, it will be evident that 
the direction P T of the resultant 
force will be always that of a 
tangent to the small cuspidated 
curve a d at T, to which tangent 
the surface of the ocean at P must 
everywhere be perpendicular, by 
reason of that law of hydrostatics 
which requires the direction of gravity to be every wnere per- 
pendicular to the surface of a fluid in equilibrio. The form 
of the curve D P A, to which the surface of the ocean will 
tend to conform itself, so as to place itself everywhere in 
equilibrio under two acting forces, will be that which always 
has P T for its radius of curvature. It will therefore be 
slightly less curved at D, and more so at A, being in £act 
no other than an ellipse, having S for its centre, d a for its 
evolute, and S A, S D for its longer and shorter semiaxes 
respectively ; so that the whole surface (supposing it covered 
with water) will tend to assume, as its form of equilibrium, 
that of an oblongated ellipsoid, having its longer axis 




6 According to Newton's calculation, the maximum disturbing force of the 
sun on the water does not exceed one 25736400th part of its gravity. That of 
the moon will therefore be to this fraction as the cube of the sun's distance 
to that of the moon's directly, and as the mass of the sun to that of the moon 
inversely, i.e. as (400) 3 x 0*011364 : 354936, which, reduced to numbers, gives, 
for the moon's maximum of power to disturb the waters, about one 12560000th 
of gravity, or somewhat less than 2£ times the sun's. 



654 OUTLINES OF ASTRONOMY 

directed toward the disturbing body, and its shorter of 
course at right angles to that direction. The difference 
of the longer and shorter semiaxes of this ellipsoid due 
to the moon's attraction would be about 58 inches: that of 
the ellipsoid, similarly formed in virtue of the sun, about 
2i times less, or about 23 inches. 

(751.) Let us suppose the moon only to act, and to have 
no orbitual motion; then if the earth also had no diurnal 
motion, the ellipsoid of equilibrium would be quietly 
formed, and all would be thenceforward tranquil. There 
is never time, however, for this spheroid to be fully formed. 
Before the waters can take their level, the moon has ad- 
vanced in her orbit, both diurnal and monthly (for in this 
theory it will answer the purpose of clearness better, if we 
suppose the earth's diurnal motion transferred to the sun 
and moon in the contrary direction), the vertex of the 
spheroid has shifted on the earth's surface, and the ocean 
has to seek a new bearing. The effect is to produce an im- 
mensely broad and excessively flat wave (not a circulating 
current), which follows, or endeavors to follow, the apparent 
motions of the moon, and must, in fact, by the principle of 
forced vibrations, imitate, by equal though not by synchro- 
nous periods, all the periodical inequalities of that motion. 
When the higher or lower parts of this wave strike our 
coasts, they experience what we call high and low water. 

(752.) The sun also produces precisely such a wave, 
whose vertex tends to follow the apparent motion of the 
sun in the heavens, and also to imitate its periodic inequal- 
ities. This solar wave co-exists with the lunar — is some- 
times superposed on it, sometimes transverse to it, so as 
to partly neutralize it, according to the monthly synodical 
configuration of the two luminaries. This alternate mutual 



OUTLINES OF ASTRONOMY 655 

reinforcement and destruction of the solar and lunar tides 
cause what are called the spring and neap tides — the former 
being their sum, the latter their difference. Although the 
real amount of either tide is, at present, hardly within 
the reach of exact calculation, yet their proportion at any 
one place is probably not very remote from that of the 
ellipticities which would belong to their respective sphe- 
roids, could an equilibrium be attained. Now these ellip- 
ticities, for the solar and lunar spheroids, are respectively 
about two and five feet; so that the average spring tide 
will be to the neap as 7 to 3, or thereabout. 

(753.) Another effect of the combination of the solar and 
lunar tides is what is called the priming and lagging of the 
tides. If the moon alone existed, and moved in the plane 
of the equator, the tide-day (i.e. the interval between two 
successive arrivals at the same place of the same vertex of 
the tide-wave) would be the lunar day (art. 143), formed 
by the combination of the moon's sidereal period and that 
of the earth's diurnal motion. Similarly, did the sun alone 
exist, and move always on the equator, the tide-day would 
be the mean solar day. The actual tide- day, then, or the 
interval of the occurrence of two successive maxima of their 
superposed waves, will vary as the separate waves approach 
to or recede from coincidence; because, when the vertices 
of two waves do not coincide, their joint height has its 
maximum at a point intermediate between them. This 
variation from uniformity in the lengths of successive tide- 
days is particularly to be remarked about the time of the 
new and full moon. 

(754.) Quite different in its origin is that deviation of 
the time of high and low water at any port or harbor, from 
the culmination of the luminaries, or of the theoretical 



656 OUTLINES OF ASTRONOMY 

maximum of their superposed spheroids, which is called 
the "establishment" of that port. If the water were with- 
out inertia, and free from obstruction, either owing to the 
friction of the bed of the sea, the narrowness of channels 
along which the wave has to travel before reaching the 
port, their length, etc., the times above distinguished would 
be identical. But all these causes tend to create a differ- 
ence, and to make that difference not alike at all ports. 
The observation of the establishments of harbors is a point 
of great maritime importance; nor is it of less consequence, 
theoretically speaking, to a knowledge of the true distribu- 
tion of the tide-waves over the globe. In making such 
observations, care must be taken not to confound the time 
of "slack water," when the current caused by the tide ceases 
to flow visibly one way or the other, and that of high or low 
water, when the level of the surface ceases to rise or fall. 
These are totally distinct phenomena, and depend on en- 
tirely different causes, though in land-locked places they 
may sometimes coincide in point of time. They are, it is 
feared, too often mistaken one for the other by practical 
men; a circumstance which, whenever it occurs, must 
produce the greatest confusion in any attempt to reduce 
the system of the tides to distinct and intelligible laws. 
(755.) The declination of the sun and moon materially 
affects the tides at any particular spot. As the vertex of 
the tide- wave tends to place itself vertically under the 
luminary which produces it, when this vertical changes its 
point of incidence on the surface, the tide-wave must tend 
to shift accordingly, and thus, by monthly and annual 
periods, must tend to increase and diminish alternately the 
principal tides. The period of the moon's nodes is thus 
introduced into this subject; her excursions in declination 



OUTLINES OF ASTRONOMY 657 

m one part of that period being 29°, and in another only 
17°, on either side the equator. 

(756.) Geometry demonstrates that the efficacy of a lumi- 
nary in raising tides is inversely proportional to the cube 
of its distance. The sun and moon, however, by reason of 
the ellipticity of their orbits, are alternately nearer to and 
further from the earth than their mean distances. In conse- 
quence of this, the efficacy of the sun will fluctuate between 
the extremes 19 and 21, taking 20 for its mean value, and 
that of the moon between 43 and 59. Taking into account 
this cause of difference, the highest spring tide will be to 
the lowest neap as 59+2-1 to 43-19, or as 80 to 24, or 10 
to 3. Of all the causes of differences in the height of tides, 
however, local situation is the most influential. In some 
places the tide-wave, rushing up a narrow channel, is sud- 
denly raised to an extraordinary height. At Annapolis, 
for instance, in the Bay of Fundy, it is said to rise 120 
feet. Even at Bristol the difference of high and low water 
occasionally amounts to 50 feet. 

(757.) It is by means of the perturbations of the planets, 
as ascertained by observation and compared with theory, 
that we arrive at a knowledge of the masses of those planets 
which having no satellites, offer no other hold upon them 
for this purpose. Every planet produces an amount of per- 
turbation in the motions of every other, proportioned to its 
mass, and to the degree of advantage or 'purchase which its 
situation in the system gives it over their movements. The 
latter is a subject of exact calculation; the former is un- 
known, otherwise than by observation of its effects. In the 
determination, however, of the masses of the planets by this 
means, theory lends the greatest assistance to observation, 
by pointing out the combinations most favorable for elicit- 



658 OUTLINES OF ASTRONOMY 

ing this knowledge from the confused mass of superposed 
inequalities which affect every observed place of a planet; 
by pointing out the laws of each inequality in its periodical 
rise and decay; and by showing how every particular in- 
equality depends for its magnitude on the mass producing 
it. It is thus that the mass of Jupiter itself (employed by 
Laplace in his investigations, and interwoven with all the 
planetary tables) has been ascertained, by observations of 
the derangements produced by it in the motions of the ultra- 
zodiacal planets, to have been insufficiently determined, or 
rather considerably mistaken, by relying too much on obser- 
vations of its satellites, made long ago by Pound and others, 
with inadequate instrumental means. The same conclusion 
has been arrived at, and nearly the same mass obtained, by 
means of the perturbations produced by Jupiter on Encke's 
comet. The error was one of great importance; the mass 
of Jupiter being by far the most influential element in the 
planetary system, after that of the sun. It is satisfactory, 
then, to have ascertained, as Mr. Airy has done, the cause 
of the error; to have traced it up to its source, in insufficient 
micrometric measurements of the greatest elongations of the 
satellites; and to have found it disappear when measures, 
taken with more care and with infinitely superior instru- 
ments, are substituted for those before employed. 

(758.) In the same way that the perturbations of the plan- 
ets lead us to a knowledge of their masses, as compared 
with that of the sun, so the perturbations of the satellites of 
Jupiter have led, and those of Saturn's attendants will no 
doubt hereafter lead, to a knowledge of the proportion their 
masses bear to their respective primaries. The system of 
Jupiter's satellites has been elaborately treated by Laplace; 
and it is from his theory, compared with innumerable obser- 



OUTLINES OF ASTRONOMY 

vations of their eclipses, that the masses assigned to them, 
in art. 540, have been fixed. Few results of theory are 
more surprising than to see these minute atoms weighed in 
the same balance, which we have applied to the ponderous 
mass of the sun, which exceeds the least of them in the 
enormous proportion of 65000000 to 1. 

(759.) The mass of the moon is concluded, 1st, from the 
proportion of the lunar to the solar tide, as observed at 
various stations, the effects being separated from each other 
by a long series of observations of the relative heights of 
spring and neap tides which, we have seen (art. 752), depend 
on the proportional influence of the two luminaries. 2dly, 
from the phenomenon of nutation, which, being the result 
of the moon's attraction alone, affords a means of calculat- 
ing her mass, independent of any knowledge of the sun's. 
Both methods agree in assigning to our satellite a mass 
about one seventy -fifth that of the earth. 6 

(760.) Not only, however, has a knowledge of the pertur- 
bations produced on other bodies of our system enabled us 
to estimate the mass of a disturbing body already known to 
exist, and to produce disturbance. It has done much more, 
and enabled geometers to satisfy themselves of the exist- 
ence, and even to indicate the situation of a planet previ- 
ously unknown, with such precision, as to lead to its im- 
mediate discovery on the very first occasion of pointing a 
telescope to the place indicated. We have already (art. 506) 
had occasion to mention in general terms this great discov- 
ery; but its importance, and its connection with the subject 
before us, call for a more specific notice of the circumstances 



6 Laplace, Expos, du Syst. du Monde, pp. 285, 300. Later researches 
have shown that this is somewhat too large, about one 88th being the value 
at present received. 



660 OUTLINES OF ASTRONOMY 

attending it. When the regular observation of Uranus, 
consequent on its discovery in 1781, had afforded some cer- 
tain knowledge of the elements of its orbit, it became possi- 
ble to calculate backward into time past, with a view to 
ascertain whether certain stars of about the same apparent 
magnitude, observed by Flamsteed, and since reported as 
missing, might not possibly be this planet. No less than 
six ancient observations of it as a supposed star were thus 
found to have been recorded by that astronomer — one in 
1690, one in 1712, and four in 1715. On farther inquiry, it 
was also ascertained to have been observed by Bradley in 
1753, by Mayer in 1756, and no less than twelve times by 
Le Monnier, in 1750, 1764, 1768, 1769, and 1771, all the 
time without the least suspicion of its planetary nature. 
The observations, however, so made, being all circumstan- 
tially registered, and made with instruments the best that 
their respective dates admitted, were quite available for cor- 
recting the elements of the orbit, which, as will be easily 
understood, is done with so much the greater precision the 
larger the arc of the ellipse embraced by the extreme obser- 
vations employed. It was, therefore, reasonably hoped and 
expected, that, by making use of the data thus afforded, 
and duly allowing for the perturbations produced since 
1690, by Saturn, Jupiter, and the inferior planets, elliptic 
elements would be obtained, which, taken in conjunction 
with those perturbations, would represent not only all the 
observations up to the time of executing the calculations, 
but also all future observations, in as satisfactory a manner 
as those of any of the other planets are actually represented. 
This expectation, however, proved delusive. M. Bouvard, 
one of the most expert and laborious calculators of whom 
astronomy has had to boast, and to whose zeal and indefati- 



OUTLINES OF ASTRONOMY 661 

gable industry we owe the tables of Jupiter and Saturn in 
actual use, having undertaken the task of constructing simi- 
lar tables for Uranus, found it impossible to reconcile the 
ancient observations above mentioned with those made from 
1781 to 1820, so as to represent both series by means of 
the same ellipse and the same system of perturbations. 
He therefore rejected altogether the ancient series, and 
grounded his computations solely on the modern, although 
evidently not without serious misgivings as to the grounds 
of such a proceeding, and "leaving it to future time to de- 
termine whether the difficulty of reconciling the two series 
arose from inaccuracy in the older observations, or whether 
it depend on some extraneous and unperceived influence 
which may have acted on the planet." 

(761.) But neither did the tables so calculated continue 
to represent, with due precision, observations subsequently 
made. The "error of the tables" after attaining a certain 
amount, by which the true longitude of Uranus was in ad- 
vance of the computed, and which advance was steadily 
maintained from about the year 1795 to 1822, began, about 
the latter epoch, rapidly to diminish, till, in 1830-31, the 
tabular and observed longitudes agreed. But, far from re- 
maining in accordance, the planet, still losing ground, fell, 
and continued to fall behind its calculated place, and that 
with such rapidity as to make it evident that the existing 
tables could no longer be received as representing, with any 
tolerable precision, the true laws of its motion. 

(762.) The reader will easily understand the nature and 
progression of these discordances by casting his eye on 
fig. 1, Plate A, in which the horizontal line or abscissa is 
divided into equal parts, each representing 50° of heliocen- 
tric longitude in the motion of Uranus round the sun, and 



662 OUTLINES OF ASTRONOMY 

in which the distances between the horizontal lines repre- 
sent each 100" of error in longitude. The result of each 
year's observation of Uranus (or of the mean of all the ob- 
servations obtained during that year) in longitude, is repre- 
sented by a black dot placed above or below the point of 
the abscissa, corresponding to the mean of the observed 
longitudes for the year: above, if the observed longitude 
be in excess of the calculated, below if it fall short of it, 
and on the line if they agree; and at a distance from the 
line corresponding to their difference on the scale above 
mentioned. 7 Thus in Flamsteed's earliest observations in 
1690, the dot so marked is placed above the line at 65 // *9 
above the line, the observed longitude being so much 
greater than the calculated. 

(763.) If, neglecting the individual points, we draw a 
curve (indicated in the figure by a fine unbroken line) 
through their general course, we shall at once perceive a 
certain regularity in its undulations. It presents two great 
elevations above, and one nearly as great intermediate de- 
pression below the medial line or abscissa. And it is evi- 
dent that these undulations would be very much reduced, 
and the errors in consequence greatly palliated, if each dot 
were removed in the vertical direction through a distance 
and in the direction indicated by the corresponding point of 
the curve A, B, C, D, E, F, G, H, intersecting the abscissa 
at points 180° distant, and making equal excursions on 
either side. Thus the point a for 1750 being removed 

7 The points are laid down from M. Leverrier's comparison of the whole 
series of observations of Uranus, with an ephemeris of his own calculation, 
founded on a complete and searching revision of the tables of Bouvard, and a 
rigorous computation of the perturbations caused by all the known planets 
capable of exercising any influence on it. The differences of longitude are 
geocentric, but for our present purpose it matters not in the least whether we 
consider the errors in heliocentric or in geocentric longitude. 



OUTLINES OF ASTRONOMY 668 

upward or in the direction toward b through a distance 
equal to c b would be brought almost to precise coincidence 
with the point d in the abscissa. Now, this is a clear indi- 
cation that a very large part of the differences in question 
is due, not to perturbation, but simply to error in the ele- 
ments of Uranus which have been assumed as the basis of 
calculation. For such excesses and defects of longitude al- 
ternating over arcs of 180° are precisely what would arise 
from error in the excentricity, or in the place of the peri- 
helion, or in both. In ellipses slightly excentric, the true 
longitude alternately exceeds and falls short of the mean 
during 180° for each deviation, and the greater the excen- 
tricity, the greater these alternate fluctuations to and fro. 
If then the excentricity of a planet's orbit be assumed er- 
roneously (suppose too great) the observed longitudes will 
exhibit a less amount of such fluctuation above and below 
the mean than the computed, and the difference of the two, 
instead of being, as it ought to be, always nil, will be alter- 
nately -f- and — over arcs of 180°. If then a difference be ob- 
served following such a law, it may arrive from erroneously 
assumed excentricity, provided always the longitudes at 
which they agree (supposed to differ by 180°) be coincident 
with those of the perihelion and aphelion; for in elliptic 
motion nearly circular, these are the points where the mean 
and true longitudes agree, so that any fluctuation of the 
nature observed, if this condition be not satisfied, cannot 
arise from error of excentricity. Now the longitude of the 
perihelion of Uranus in the elements employed by Bouvard 
is (neglecting fractions of a degree) 168°, and of the aphelion 
348°. These points then, in our figure, fall at <o and a re- 
spectively, that is to say, nearly half way between A C, 
C E, EG, etc. It is evident therefore that it is not to 



66± OUTLINES OF ASTRONOMY 

error of excentricity that the fluctuation in question is 
mainly due. 

(764.) Let us now consider the effect of an erroneous 
assumption of the place of the perihelion. Suppose in 
fig. 2, Plate A, o x to represent the longitude of a planet, 
and x y the excess of its true above its mean longitude, 
due to ellipticity. Then if K be the place of the perihelion, 
and P, or T, the aphelion in longitude, y will always lie in 
a certain undulating curve PQEST, above 8 P T between 
K and T, and below it between P and E. Now suppose the 
place of the perihelion shifted forward to r, or the whole 
curve shifted bodily forward into the situation p q r s t, then 
at the same longitude o x, the excess of the true above the 
mean longitude will be x y' only; in other words, this 
excess will have diminished by the quantity y y' below 
its former amount. Take therefore in o N (fig. 3, Plate A) 
o y=o x and y y' B.lwa.j8=y y' in fig. 2, and having thus con- 
structed the curve KLMNO, the ordinate y y' will always 
represent the effect of the supposed change of perihelion. 
It is evident (the excentricity being always supposed small) 
that this curve will consist also of alternate superior and 
inferior waves of 180° each in amplitude, and the points L, 
N of its intersection with the axis will occur at longitudes 
corresponding to X, Y intermediate between the maxima 
Q, q and S, s of the original curves, that is to say (if these 
intervals Q q, S s, or E r to which both are equal, be very 
small) very nearly at 90° from the perihelion and aphelion. 
Now this agrees with the conditions of the case in hand, 
and we are therefore authorized to conclude that the major 
portion of the errors in question has arisen from error in 
the place of the perihelion of Uranus itself, and not from 

8 The curves, figs. 2, 3, are inverted in the engraving. 



OUTLINES OF ASTRONOMY 665 

perturbation, and that to correct this portion, the perihelion 
must be shifted somewhat forward. As to the amount of 
this shifting, our only object being explanation, it will not 
be necessary here to inquire into it. It will suffice that it 
must be such as shall make the curve ABCDEFGas 
nearly as possible similar, equal, and opposite to the curve 
traced out by the dots on the other side. And this being 
done, we may next proceed to lay down a curve of the 
residual differences between observation and theory in the 
mode indicated in art. 763. 

(765.) This being done, by laying off at each point of 
the line of longitudes an ordinate equal to the difference 
of the ordinates of the two curves in fig. 1, when on op- 
posite, and their sum when on the same side of the abscissa, 
the result will be as indicated by the dots in fig. 4. And 
here it is at once seen that a still further reduction of the 
differences under consideration would result, if, instead 
of taking the line A B for the line of longitudes, a line 
a b slightly inclined to it were substituted, in which case 
the whole of the differences between observation and theory 
from 1712 to 1800 would be annihilated, or at least so far 
reduced as hardly to exceed the ordinary errors of observa- 
tion; and as respects the observation of 1690, the still out- 
standing difference of about 35" would not be more than 
might be attributed to a not very careful observation at so 
early an epoch. JSTow the assumption of such a new line of 
longitudes as the correct one is in effect equivalent to the 
admission of a slight amount of error in the periodic time 
and epoch of Uranus; for it is evident that by reckoning 
from the inclined instead of the horizontal line, we in effect 
alter all the apparent outstanding errors by an amount pro- 
portional to the time before or after the date at which the 



OUTLINES OF ASTRONOMY 

two lines intersect (viz. about 1789). As to the direction 
in which this correction should be made, it is obvious by 
inspection of the course of the dots, that if we reckon from 
A B, or any line parallel to it, the observed planet on the 
long run keeps falling more and more behind the calculated 
one; i.e. its assigned mean angular velocity by the tables 
is too great and must be diminished, or its periodic time 
requires to be increased. 

(766.) Let this increase of period be made, and in corre- 
spondence with that change let the longitudes be reckoned 
on a b, and the residual differences from that line instead 
of A B, and we shall have then done all that can be done 
in the way of reducing and palliating these differences, and 
that, with such success, that up to the year 1804 it might 
have been safely asserted that positively no ground what- 
ever existed for suspecting any disturbing influence. But 
with this epoch an action appears to have commenced, and 
gone on increasing, producing an acceleration of the motion 
in longitude, in consequence of which Uranus continually 
gains on its elliptic place, and continued to do so till 1822, 
when it ceased to gain, and the excess of longitude was at 
its maximum, after which it began rapidly to lose ground, 
and has continued to do so up to the present time. It is 
perfectly clear, then, that in this interval some extraneous 
cause must have come into action which was not so before, 
or not in sufficient power to manifest itself by any marked 
effect, and that that cause must have ceased to act, or rather 
begun to reverse its action, in or about the year 1822, the 
reverse action being even more energetic than the direct. 

(767.) Such is the phenomenon in the simplest form we 
are now able to present it. Of the various hypotheses 
formed to account for it, during the progress of its develop- 



OUTLINES OF ASTRONOMY 667 

ment, none seemed to have any degree of rational proba- 
bility but that of the existence of an exterior, and hitherto 
undiscovered, planet, disturbing, according to the received 
laws of planetary disturbance, the motion of Uranus by its 
attraction, or rather superposing its disturbance on those 
produced by Jupiter and Saturn, the only two of the old 
planets which exercise any sensible disturbing action on 
that planet. Accordingly, this was the explanation which 
naturally , and almost of necessity, suggested itself to those 
conversant with the planetary perturbations who considered 
the subject with any degree of attention. The idea, how- 
ever, of setting out from the observed anomalous deviations, 
and employing them as data to ascertain the distance and 
situation of the unknown body, or, in other words, to re- 
solve the inverse problem of perturbations, '"'given the dis- 
turbances to find the orbit, and place in that orbit of the 
disturbing planet," appears to have occurred only to two 
mathematicians, Mr. Adams in England and M. Leverrier 
in France, with sufficient distinctness and hopefulness of 
success to induce them to attempt its solution. Botb suc- 
ceeded, and their solutions, arrived at with perfect inde- 
pendence, and by each in entire ignorance of the other's 
attempt, were found to agree in a surprising manner when 
the nature and difficulty of the problem is considered; the 
calculations of M. Leverrier assigning for the heliocentric 
longitude of the disturbing planet for the 23d September, 
1846, 326° 0', and those of Mr. Adams (brought to the same 
date) 329° 19', differing only 3° 19'; the plane of its orbit 
deviating very slightly, if at all, from that of the ecliptic. 
(768.) On the day above mentioned — a day forever 
memorable in the annals of astronomy — Dr. Gralle, one of 
the astronomers of the Royal Observatory at Berlin, re- 



OUTLINES OF ASTRONOMY 

ceived a letter from M. Leverrier, announcing to him the 
result he had arrived at, and requesting him to look for 
the disturbing planet in or near the place assigned by his 
calculation. He did so, and on that very night actually 
found it. A star of the eighth magnitude was seen by him 
and by M. Encke in a situation where no star was marked 
as existing in Dr. Bremiker's chart, then recently published 
by the Berlin Academy. The next night it was found to 
have moved from its place, and was therefore assuredly 
a planet. Subsequent observations and calculations have 
fully demonstrated this planet, to which the name of Nep- 
tune has been assigned, to be really that body to whose 
disturbing attraction, according to the Newtonian law of 
gravity, the observed anomalies in the motion of Uranus 
were owing. The geocentric longitude determined by Dr. 
Gralle from this observation was 325° 53', which, converted 
into heliocentric, gives 326° 52', differing 0° 52' from M. 
Leverrier's place, 2° 27' from that of Mr. Adams, and only 
47' from a mean of the two calculations. 

(769.) It would be quite beyond the scope of this work, 
and far in advance of the amount of mathematical knowl- 
edge we have assumed our readers to possess, to attempt 
giving more than a superficial idea of the course followed 
by these geometers in their arduous investigations. Suffice 
it to say, that it consisted in regarding, as unknown quan- 
tities, to be determined, the mass, and all the elements of 
the unknown planet (supposed to revolve in the same 
plane and the same direction with Uranus), except its 
major semiaxis. This was assumed in the first instance 
(in conformity with "Bode's law," art. 505, and certainly 
at the time with a high prima facie probability) to be 
double that of Uranus, or 38*364 radii of the earth's orbit. 



OUTLINES OF ASTRONOMY 669 

Without some assumption as to the value of this element, 
owing to the peculiar form of the analytical expression of 
the perturbations, the analytical investigation would have 
presented difficulties apparently insuperable. But besides 
these, it was also necessary to regard as unknown, or at 
least as liable to corrections of unknown magnitude of the 
same order as the perturbations, all the elements of Uranus 
itself, a circumstance whose necessity will easily be under- 
stood, when we consider that the received elements could 
only be regarded as provisional, and must certainly be 
erroneous, the places from which they were obtained being 
affected by at least some portions of the very perturba- 
tions in question. This consideration, though indispensa- 
ble, added vastly both to the complication and the labor 
of the inquiry. The axis (and therefore the mean mo- 
tion) of the one orbit, then, being known very nearly, and 
that of the other thus hypothetically assumed, it became 
practicable to express in terms, partly algebraic, partly nu- 
merical, the amount of perturbation at any instant, by 
the aid of general expressions delivered by Laplace in his 
"Mecanique Celeste 11 and elsewhere. These, then, together 
with the corrections due to the altered elements of Uranus 
itself, being applied to the tabular longitudes, furnished, 
when compared with those observed, a series of equations, 
in which the elements and mass of Neptune, and the correc- 
tions of those of Uranus entered as the unknown quantities, 
and by whose resolution (no slight effort of analytical skill) 
all their values were at length obtained. The calculations 
were then repeated, reducing at the same time the value of 
the assumed distance of the new planet, the discordances 
between the given and calculated results indicating it to 
have been assumed too large; when the results were found 



670 



OUTLINES OF ASTRONOMY 



to agree better, and the solutions to be, in fact, more satis- 
factory. Thus, at length, elements were arrived at for the 
orbit of the unknown planet, as below. 



Epoch of Elements 
Mean longitude in Epoch 
Semiaxis Major 
Excentricity . 
Longitude of Perihelion 
Mass (the Sun being 1) . 



Leverrier. 



Jan. 1, 1847. 
318° 47' 4 
36-1539 
0.107610 
284° 45' 8 
0-00010727 



Adams. 



Oct. 6, 1846. 

323° 2' 

37-2474 

0-120615 

299° II' 

0-00015003 



The elements of M. Leverrier were obtained from a con- 
sideration of the observations up to the year 1845, those of 
Mr. Adams, only as far as 1840. On subsequently taking 
into account, however, those of the five years up to 1845, 
the latter was led to conclude that the semiaxis ought to 
be still much further diminished, and that a mean distance 
of 33*33 (being to that of Uranus as 1 : 0*574) would prob- 
ably satisfy all the observations very nearly. 9 

(770.) On the actual discovery of the planet, it was, of 
course, assiduously observed, and it was soon ascertained 
that a mean distance, even less than Mr. Adams's last pre- 
sumed value, agreed better with its motion; and no sooner 
were elements obtained from direct observation, sufficiently 
approximate to trace back its path in the heavens for a con- 
siderable interval of time, than it was ascertained to have 
been observed as a star by Lalande on the 8th and 10th 
of May, 1795, the latter of the two observations, however, 
having been rejected by him as faulty, by reason of its non- 
agreement with the former (a consequence of the motion 
of the planet in the interval). From these observations, 



9 In a letter to the Astronomer Royal, dated Sept. 2, 1846 — i.e. three weeks 
previous to the optical discovery of the planet. 



OUTLINES OF ASTRONOMY 671 

combined with those since accumulated, the elements cal- 
culated by Prof. Walker, U. S., result as follows: 



Epoch of Elements 

Mean longitude at Epoch 

Semiaxis Major . 

Excentricity 

Longitude of the Perihelion 

Ascending Node 

Inclination . 

Periodic time 

Mean annual Motion . 



Jan. 1, 1847, M. noon, Greenwich 
328° 32' 44" -2 
30-0367 

0-00871946 

47° 12' 6"-50 

130° 4' 20"-81 

1° 46' 58" -97 
164-6181 tropical year 
2°-18688 



(771.) The great disagreement between these elements 
and those assigned either by M. Leverrier or Mr. Adams 
will not fail to be remarked ; and it will naturally be asked 
how it has come to pass, that elements so widely different 
from the truth should afford anything like a satisfactory 
representation of the perturbation in question, and that the 
true situation of the planet in the heavens should have been 
so well, and indeed accurately, pointed out by them. As 
to the latter point, any one may satisfy himself by half an 
hour's calculation that both sets of elements do really place 
the planet, on the day of its discovery, not only in the 
longitudes assigned in art. 763, i.e. extremely near its 
apparent place, but also at a distance from the Sun very 
much more approximately correct than the mean distances 
or semiaxes of the respective orbits. Thus the radius vector 
of Neptune, calculated from M. Leverrier's elements for 
the day in question, instead of 36-1539 (the mean distance) 
comes out almost exactly 33; and indeed, if we consider 
that the excentricity assigned by those elements gives for 
the perihelion distance 32-2634, the longitude assigned to 
the perihelion brings the whole arc of the orbit (more than 
83°), described in the interval from 1806 to 1847 to lie 
within 42° one way or the other of the perihelion, and 
therefore, during the whole of that interval, the hypotheti- 



672 



OUTLINES OF ASTRONOMY 



cal planet would be moving within limits of distance from 
the sun, 32*6 and 83*0. The following comparative tables 
of the relative situations of Uranus, the real «nd hypo- 
thetical planet, will exhibit more clearly than any length- 
ened statement, the near imitation of the motion of the 
former by the latter within that interval. The longitudes 
are heliocentric. 10 



A. D. 


Uranus. 


Neptune. 


Leverrier. 


Adams. 


Long. 


Long. 


Rad. Vec. 


Long. 


Rad. Vec. 


Long. 


Rad. Vec. 


1805-0 


197°-8 


235°-9 


30-3 


241°-2 


33-1 


246°-5 


34-2 


1810-0 


220-9 


247-0 


30-3 


251-1 


32-8 


255-9 


33-7 


1815-0 


243-2 


258-0 


30-3 


261-2 


32-5 


265-5 


33-3 


1820-0 


264-7 


268-8 


30-2 


271-4 


32-4 


275-4 


331 


1821-0 


269-0 


271-0 


30-2 


273-5 


32-3 - 


277-4 


33-0 


1822-0 


273-3 


273-2 


30-2 


275-6 


32-3 


279-5 


33-0 


1823-0 


277-6 


275-3 


30-2 


277-6 


32-3 


281-5 


32-9 


1824-0 


281-8 


277-4 


30-2 


279-7 


32-3 


283-6 


32-9 


1825-0 


285-8 


279-6 


30-2 


281-8 


32 3 


285-6 


32-8 


1830-0 


306-1 


290-5 


30-1 


292-1 


32-3 


296-0 


32-8 


1835-0 


326-0 


301-4 


30-1 


302-5 


32-4 


306-3 


32-8 


1840-0 


345-7 


312-2 


30-1 


312-6 


32-6 


316-3 


32-9 


1845-0 


365-3 


323-1 


30-0 


322-6 


32-9 


326-0 


33-1 


1847-0 


373-3 


327-6 


30-0 


326-5 


33-1 


329-3 


33-2 



(772.) From this comparison it will be seen that Uranus 
arrived at its conjunction with Neptune at or immediately 
before the commencement of 1822, with the calculated 
planet of Leverrier at the beginning of the following year 
1823, and with that of Adams about the end of 1824. Both 
the theoretical planets, and especially that of M. Leverrier, 
therefore, during the whole of the above interval of time, 
so far as the directions of their attractive forces on Uranus 
are concerned, would act nearly on it as the true planet must 
have done. As regards the intensity of the relative dis- 
turbing forces, if we estimate these by the principles of art, 



10 The calculations are carried only to tenths of degrees, as quite sufficient 
for the object in view. 



OUTLINES OF ASTRONOMY 673 

612 at the epochs of con j auction, and for the commence- 
ment of 1805 and 18-15, we find for the respective denomi- 
nators of the fractions of the sun's attraction on Uranus 
regarded as unity, which express the total disturbing force, 
N S, in each case, as below: 

1805. Conjunction. 1845. 

J Peirce ? s mass r^r^ 2*7540 7508 32390 

Neptune with -l 

Struve' s mass — -- 20244 5519 23810 

[_ 14496 

1 

Leverrier's theoretical Planet, mass r^r 20837 5193 19935 

The masses here assigned to Neptune are those respectively 
deduced by Prof. Peirce and M. Struve from observations 
of the satellite discovered by Mr. Lassell, made with the 
large telescopes of Fraunhofer in the observatories of Cam- 
bridge, U. S., and Pulkova respectively. These it will be 
perceived differ very considerably, as might reasonably be 
expected in the results of micrometrical measurements of 
such difficulty, and it is not possible at present to say 
to which the preference ought to be given. As compared 
with the mass assigned by M. Struve, an agreement on the 
whole more satisfactory could not have been looked for 
within the interval immediately in question. 

(773.) Subject then to this uncertainty as to the real 
mass of Neptune, the theoretical planet of Leverrier must 
be considered as representing with quite as much fidelity as 
could possibly be expected in a research of such exceeding 
delicacy, the particulars of its motion and perturbative ac- 
tion during the forty years elapsed from 1805 to 1845, an 
interval which (as is obvious from the rapid diminution of 
the forces on either side of the conjunction indicated by the 
numbers here set down) comprises all the most influential 



674 OUTLINES OF ASTRONOMY 

range of its action. This will, however, be placed in full 
evidence by the construction of curves representing the nor- 
mal and tangential forces on the principles laid down (as 
far as the normal constituent is concerned) in art. 717, one 
slight change only being made, which, for the purpose in 
view, conduces greatly to clearness of conception. The 
force L s (in the figure of that article) being supposed ap- 
plied at P in the direction L s, we here construct the curve 
of the normal force by erectmg at P (fig. 5, Plate A) P W 
always perpendicular to the disturbed orbit, A P, at P, 
measured from P in the same direction that S lies from L, 
and equal in length to L S. P W then will always repre- 
sent both the direction and magnitude of the normal force 
acting at P. And in like manner, if we take always P Z on 
the tangent to the disturbed orbit at P, equal to N L of the 
former figure, and measured in the same direction from P 
that L is from N, PZ will represent both in magnitude and 
direction the tangential force acting at P. Thus will be 
traced out the two curious ovals represented in our figure of 
their proper forms and proportions for the case in question. 
That expressing the normal force is formed of four lobes, 
having a common point in S, viz. SW m XSaSnSJ 
S W, and that expressing the tangential, A Z c f B e d 
Y A Z, consisting of four mutually intersecting loops, sur- 
rounding and touching the disturbed orbit in four points, 
A B c d. The normal force acts outward over all that part 
of the orbit, both in conjunction and opposition, correspond- 
ing to the portions of the lobes m, n, exterior to the dis- 
turbed orbit, and inward in every other part. The figure 
sets in a clear light the great disproportion between the 
energy of this force near the conjunction, and in any other 
configuration of the planets ; its exceedingly rapid degrada- 



OUTLINES OF ASTRONOMY 675 

tion as P approaches the point of neutrality (whose situation 
is Sb° 5' on either side of the conjunction, an arc described 
synodically by Uranus in 16 y *72); and the comparatively 
short duration and consequent inefficacy to produce any 
great amount of perturbation, of the more intense part of 
its inward action in the small portions of the orbit corre- 
sponding to the lobes «, b, in which the line representing 
the inward force exceeds the radius of the circle. It ex- 
hibits, too, with do less distinctness, the gradual develop- 
ment, and rapid degradation and extinction of the tangential 
force from its neutral points, c, d, on either side up to the 
conjunction, where its action is reversed, being accelerative 
over the arc d A, and retardative over A c, each of which 
arcs has an amplitude of 71° 20', and is described by Uranus 
synodically in 34 y, 00. The insignificance of the tangential 
force in the configurations remote from conjunction through- 
out the arc c B d is also obviously expressed by the small 
comparative development of the loops e, f. 

(774.) Let us now consider how the action of these forces 
results in the production of that peculiar character of per- 
turbation which is exhibited in our curve, fig. 4, Plate A. 
It is at once evident that the increase of the longitude from 
1800 to 1822, the cessation of that increase in 1822, and its 
conversion into a decrease during the subsequent interval is 
in complete accordance with the growth, rapid decay, ex- 
tinction at conjunction, and subsequent reproduction in a 
reversed sense of the tangential force: so that we cannot 
hesitate in attributing the greater part of the perturbation 
expressed by the swell and subsidence of the curve between 
the years 1800 and 1845 — all that part, indeed, which is 
symmetrical on either side of 1822 — to the action of the 

tangential force. 

Astronomy — Vol. XX — 10 



676 OUTLINES OF ASTRONOMY 

(775.) But it will be asked — has then the normal force 
(which, on the plain showing of fig. 5, is nearly twice as 
powerful as the tangential, and which does not reverse its 
action, like the latter force, at the point of conjunction, but, 
on the contrary, is there most energetic) no influence in pro- 
ducing the observed effects ? "We answer, very little, within 
the period to which observation had extended up to 1845. 
The effect of the tangential force on the longitude is direct 
and immediate (art. 660), that of the normal indirect, conse- 
quential, and cumulative with the progress of time (art. 734). 
The effect of the tangential force on the mean motion takes 
place through the medium of the change it produces on the 
axis, and is transient: the reversed action after conjunction 
(supposing the orbits circular) exactly destroying all the 
previous effect, and leaving the mean motion on the whole 
unaffected. In the passage through the conjunction, then, 
the tangential force produces a sudden and powerful accel- 
eration, succeeded by an equally powerful and equally sud- 
den retardation, which done, its action is completed, and no 
trace remains in the subsequent motion of the planet that it 
ever existed, for its action on the perihelion and excentricity 
is in like manner also nullified by its reversal of direction. 
But with the normal force the case is far otherwise. Its im- 
mediate effect on the angular motion is nil. It is not till it 
has acted long enough to produce a perceptible change in 
the distance of the disturbed planet from the sun that the 
angular velocity begins to be sensibly affected, and it is not 
till its whole outward action has been exerted (i.e. over the 
whole interval from neutral point to neutral point) that its 
maximum effect in lifting the disturbed planet away from 
the sun has been produced, and the full amount of diminu- 
tion in angular velocity it is capable of causing has been de- 



OUTLINES OF ASTRONOMY 677 

veloped. This continues to act in producing a retardation 
in longitude long after the normal force itself has reversed 
its action, and from a powerful outward force has become a 
feeble inward one. A certain portion of this perturbation is 
incident on the epoch in the mode described in art. 731 et 
seq., and permanently disturbs the mean motion from what 
it would have been, had Neptune no existence. The rest of 
its effect is compensated in a single synodic revolution, not 
by the reversal of the action of the force (for that reversed 
action is far too feeble for this purpose), but by the effect of 
the permanent alteration produced in the excentricity, which 
(the axis being unchanged) compensates by increased prox- 
imity in one part of the revolution, for increased distance 
in the other. Sufficient time has not yet elapsed since the 
conjunction to bring out into full evidence the influence of 
this force. Still its commencement is quite unequivocally 
marked in the more rapid descent of our curve fig. 4, sub- 
sequent to the conjunction than ascent previous to that 
epoch, which indicates the commencement of a series of 
undulations in its future course of an elliptic character, con- 
sequent on the altered excentricity and perihelion (the total 
and ultimate effect of this constituent of the disturbing 
force) which will be maintained till within about 20 years 
from the next conjunction, with the exception, perhaps, of 
some trifling inequalities about the time of the opposition, 
similar in character, but far inferior in magnitude to those 
now under discussion. 

(776.) Posterity will hardly credit that, with a full knowl- 
edge of all the circumstances attending this great discovery 
— of the calculations of Leverrier and Adams — of the com- 
munication of its predicted place to Dr. Galle — and of the 
new planet being actually found by him in that place, in the 



678 OUTLINES OF ASTRONOMY 

remarkable manner above commemorated; Dot only have 
doubts been expressed as to the validity of the calculations 
of those geometers, and the legitimacy of their conclusions, 
but these doubts have been carried so far as to lead the ob- 
jectors to attribute the acknowledged fact of a planet previ- 
ously unknown occupying that precise place in the heavens 
at that precise time, to sheer accident! J1 What share acci- 
dent may have had in the successful issue of the calcula- 
tions, we presume the reader, after what has been said, will 
have little difficulty in satisfying himself. As regards the 
time when the discovery was made, much has also been at- 
tributed to fortunate coincidence. The following considera- 
tions will, we apprehend, completely dissipate this idea, if 
still lingering in the mind of any one at all conversant with 
the subject. The period of Uranus being 84-0140 years, and 
that of Neptune 164-6181, their synodic revolution (art. 418), 
or the interval between two successive conjunctions, is 

11 These doubts seem to have originated partly in the great disagreement 
between the predicted and real elements of Neptune, partly in the near {possibly 
precise) commensurability of the mean motions of Neptune and Uranus. We 
conceive them however to be founded in a total misconception of the nature of 
the problem, which was not, from such obviously uncertain indications as the 
observed discordances could give, to determine as astronomical quantities the 
axis, excentricity and mass of the disturbing planet; but practically to discover 
where to look for it; when, if once found, these elements would be far better 
ascertained. To do this, any axis, excentricity , 'perihelion and mass, however 
vjide of the truth, which would represent, even roughly, the amount, but with 
tolerable correctness the direction of the disturbing force during the very moder- 
ate interval when the departures from theory were really considerable, would 
equally serve their purposes; and with an excentricity, mass and perihelion 
disposable, it is obvious that any assumption of the axis between the limits 30 
and 38, nay, even with a much wider inferior limit, would serve the purpose. 
In his attempt to assign an inferior limit to the axis, and in the value so as- 
signed, M. Leverrier, it must be admitted, was not successful. Mr. Adams, 
on the other hand, influenced by no considerations of the kind which appear 
to have weighed with his brother geometer, fixed ultimately (as we have seen) 
before the actual discovery of the planet, on an axis not very egregiously wrong. 
Still it were to be wished, for the satisfaction of all parties, that some one would 
undertake the problem de novo, employing formulas not liable to the passage 
through infinity, which, technically speaking, hampers, or may be supposed to 
hamper, the continuous application of the usual perturbational formulae when 
cases of commensurability occur. 



OUTLINES OF ASTRONOMY 679 

17158 years. The late conjunction having taken place 
about the beginning of 1822; that next preceding must 
have happened in 1649, or more than 40 years before the 
first recorded observation of Uranus in 1690, to say nothing 
of its discovery as a planet. In 1690, then, it must have 
been effectually out of reach of any perturbative influence 
worth considering, and so it remained during the whole in- 
terval from thence to 1800. From that time the effect of 
perturbation began to become sensible, about 1805 promi- 
nent, and in 1820 had nearly reached its maximum. At this 
epoch an alarm was sounded. The maximum was not at- 
tained — the event, so important to astronomy, was still in 
progress of development — when the fact (anything rather 
than a striking one) was noticed, and made matter of com- 
plaint. But the time for discussing its cause with any pros- 
pect of success was not yet come. Everything turns upon 
the precise determination of the epoch of the maximum, 
when the perturbing and perturbed planet were in conjunc- 
tion, and upon the law of increase and diminution of the per- 
turbation itself on either side of that point. Now it is 
always difficult to assign the time of the occurrence of a 
maximum by observations liable to errors bearing a ratio 
far from inconsiderable to the whole quantity observed. 
Until the lapse of some years from 1822 it would have been 
impossible to have fixed that epoch with any certainty, and 
as respects the law of degradation and total arc of longitude 
over which the sensible perturbations extend, we are hardly 
yet arrived at a period when this can be said to be com- 
pletely determinable from observation alone. In all this 
we see nothing of accident, unless it be accidental that an 
event which must have happened between 1781 and 1953, 
actually happened in 1822; and that we live in an age when 



680 OUTLINES OF ASTRONOMY 

astronomy has reached that perfection, and its cultivators 
exercise that vigilance which neither permit such an event, 
nor its scientific importance, to pass unnoticed. The blos- 
som had been watched with interest in its development, and 
the fruit was gathered in the very moment of maturity. 12 

(776 a.) In the foregoing chapters we have enumerated 
and described the several bodies so far as known of which 
our system consists, and have shown how their mutual dis- 
tances from and their motions with respect to each other 
may be determined, and their masses compared with that of 
the central body, and ultimately with that of our own planet 
as a unit of reference; but nothing has been said respecting 
the means by which that unit itself can be brought into 
comparison with the mass, weight or inertia of those por- 
tions of its substance which we see and handle on its surface. 
This datum — the total weight of the earth itself — the num- 
ber of times that its entire mass exceeds that of a pound of 
lead or other matter — or in other words (its bulk being ac- 
curately known) its mean density — remains up to this point 
of the present work undetermined, and is the one thing 
wanting to complete our knowledge of the data of our sys- 
tem and fully to connect astronomy with ordinary mechan- 
ics. We shall now therefore proceed to explain the meth- 
ods by which this has been accomplished. 

12 The student who may wish to see the perturbations of Uranus produced 
by Neptune, as computed from a knowledge of the elements and maas of that 
planet, such as we now know to be pretty near the truth, will find them stated 
at length from the calculations of Mr. Walker (of Washington, U. S.), in the 
"Proceedings of the American Academy of Art3 and Sciences," vol. i. p. 334 
et seq. On examining the comparisons of the results of Mr. Walker's formulas 
with those of Mr. Adams's theory in p. 342, he will perhaps be surprised at 
the enormous difference between the actions of Neptune and Mr. Adams's 
"hypothetical planet" on the longitude of Uranus. This is easily explained. 
Mr. Adams's perturbations are deviations from Bouvard's orbit of Uranus, as 
it stood immediately previous to the late conjunction. Mr. Walker's are the 
deviations from a mean or undisturbed orbit freed from the influence of the long 
inequality resulting from the near commensurability of the motions. 



OUTLINES OF ASTRONOMY 681 

(770 b.) The principle which at once suggests itself to 
everv mind is to measure the direct attraction, if it be pos- 
sible, of some known mass, at some known distance, on 
some other. We say, if it be possible, because whatever 
notion we may form d priori of the weight of the earth as 
estimated in pounds or tons, it is clearly something enor- 
mous; and moreover, since it follows from the law of gravi- 
tative attraction 13 that the attractions of spheres of equal 
density on points at their surface are to each other as their 
radii, the attraction of a globe a foot in diameter, of the 
same average density of the earth, on a material point at its 
own surface would only amount to the 41,849,280th part of 
the weight of such material point; and therefore its attrac- 
tion on a spherical body, suppose also a foot in diameter, 
placed in contact with it, would only amount to one 167,- 
397,120th part of the weight of such body. Now when we 
have to deal with fractions of such an order of minuteness, 
all ordinary modes of directly measuring forces and weights 
break down, and the utmost resources of invention and art 
must be taxed even to render them perceptible, to say noth- 
ing of their precise determination. 

(776 c.) The first and most obvious mode of producing a 
magnified result is to augment, in as high a ratio as possible, 
the attracting mass; and therefore to substitute some great 
natural mass of the most suitable form which can be found, 
for an artificial sphere. And as the resources afforded by 
the integral calculus furnish the means of calculating the 
attraction of a body of any size and figure of known mate- 
rials on a point anywhere situated without it, the idea natu- 
rally enough suggested itself to take some large mountain, 

13 Princip. bk. 1, prop. 72. 



682 OUTLINES OF ASTRONOMY 

of as regular a shape as might be found, for the attracting 
body, and to measure its attraction, on a principle pointed 
out by Newton, 14 by the deviation from verticality of a 
plumb-line suspended near it, which will necessarily ba 
drawn aside toward the mountain. As the deflection to bo 
expected however, even in the case of a very large moun- 
tain, is still exceedingly minute, the working out of this 
idea into practice calls for very exact and refined astronom- 
ical observations. 

(776 d.) In the first place the question arises in limine, 
how are we to ascertain, at any place, what is a vertical 
direction? The deviated plumb-line, it is obvious, cannot 
give us this information, nor can levels, for the surface of 
still water is always at right angles to the single force, what- 
ever that may be, which results from a combination cf all 
the forces acting on it — in other words, to the direction of 
the deviated plumb-line. Here it is that our knowledge 
of the figure and dimensions of the earth stand us in stead. 
We cannot, it is true, remove the mountain so as to find 
where the plumb-line would point, or the level rest in its 
absence; but we can shift our station to the opposite side, 
and by sidereal observation ascertain whether the direction 
of the plumb-line has varied, by more or less than the 
amount of change due to such a change of station on the 
globe. Thus then we proceed: — 

Suppose M the mountain, A B a circle of latitude pass- 
ing through two stations P, Q, at its foot (or rather at such 
heights on its slope as shall correspond to the maxima of its 
lateral attraction), at each of which let observations be made 
with a portable zenith sector alternately established at one 

14 Treatise on the System of the World in a popular way (1728). 



OUTLINES OF ASTRONOMY 683 

and the other of the zenith distances of some star passing 
very near the zenith of the mountain (so as to free the obser- 
vations from uncertainty of refraction). Were there no lat- 
eral attraction, the plumb-lines at both stations would point 
directly to the centre of curvature C of the terrestrial spher- 
oid (art. 219), and the angle between them, P C Q, would 
be the difference of latitudes of the stations. Now the 
dimensions and ellipticity of the earth as a whole being 
known, this latter difference can be independently deter- 
mined by a trigonometrical survey instituted for the pur- 




pose, a base being measured, and the meridional distance 
P Q ascertained by triangulation (art. 274 et seq.), which, 
converted into seconds of latitude, gives the difference in 
question; to which, were there no local attraction, the ob- 
served difference of zenith distances ought to correspond. 
But this will not be the case. The mountain will attract 
the plummet both ways inward, into situations PR, QE, 
including a greater angle than P C Q, and this being the ob- 
served angle or apparent difference of zenith distances — sub- 
tracting from it the difference of latitude so independently 



681 OUTLINES OF ASTRONOMY 

obtained, the excess will represent the sum of the two devia- 
tions north and south due to the attraction required. The 
mountain has then to be surveyed, and modelled, and min- 
eralogical specimens taken from every accessible part of it, 
and their specific gravities determined; and thus, no matter 
with what amount of calculation (for it is no light task), the 
total lateral attraction is computed in units of some definite 
scale; such, for instance, that each unit shall represent the 
total attraction of a sphere of lib. weight, on a point 1 foot 
distant from its centre. The sum of all these units, each 
reduced to a horizontal direction, is the total lateral attrac- 
tion of the mountain, and is therefore to the total vertical 
attraction of the earth as the tangent of the deviation (taken 
so as to divide the total observed difference in the ratio of 
the computed attractions at either station) is to radius. 

(776 e.) The process is laborious and costly — requires 
excellent instruments and the co-operation of more than one 
practiced observer. It has, however, been put in execution 
on several occasions; viz., 1st, by the French Academicians, 
Bouguer and La Condamine, who, in the course of their 
operations in Peru for the measurement of an arc of the 
meridian (art. 216), instituted observations of the kind above 
described on Chimborazo in 1738. Their means of observa- 
tion, however, were not such as to afford any distinct result, 
though a deviation of the plumb-line to the amount of about 
IF appears to have been obtained. 2d, by Maskelyne, in 
1774, on the mountain Schehallien in Scotland, a mountain, 
not mdeed of any great magnitude, being only about 3000 
feet in altitude, but well situated, and otherwise well adapted 
for the experiment. It was successful. A joint amount of 
the lateral deviations on either side, of 11**6, was well ascer- 
tained to be produced by the local attraction, and the calcu- 



OUTLINES OF ASTRONOMY 685 

lations being executed (by Dr. Hutton and subsequently by 
Professor Playfair), a result entitled to some reliance was 
obtained, according to which the mean density of the earth 
comes out -i'713 times that of water at the surface. More 
recently, we find a series of observations instituted by Sir 
H. James, Superintendent of the Ordnance Survey, on Ar- 
thur's Seat near Edinburgh, 16 by which, from a deflexion of 
2" -21 observed on the north and of 2"*00 on the south side 
of that mountain, a mean density results of 5 "316. 

(776/.) Observations of the time of oscillation of a pen- 
dulum afford (see art. 235) a direct measure of the force by 
which the oscillating mass is urged vertically downward. 
Hence it follows that if this time be very precisely deter- 
mined, both at the summit and at the foot of a mountain or 
elevated tableland, the attraction of the mass of such moun- 
tain or tableland vertically downward will become known. 
For gravity decreasing inversely as the square of the dis- 
tance would be enfeebled by the increase of that distance in 
a proportion which can be precisely calculated from the 
known height of the upper station; and therefore, could 
the pendulum be supported in the air at that height, the 
increase of its time of oscillation, under those circumstances, 
would be exactly known by calculation. But being sup- 
ported on a mountain mass, protruding above the level sur- 
face of the terrestrial spheroid, the attraction of that mass 
acts on it in addition to the so diminished force of general 
gravitation, and prevents it from losing on the sea-level rate 
so much as it would do were the mountain devoid of attrac- 
tive power. Experiments of this nature have been made by 
the Italian astronomers Plana and Carlini on Mont Cenis in 

15 Phil. Tr. 1856, p. 591. 



OUTLINES OF ASTRONOMY 

Savoy, and the result, all computations executed, have given 
4*950 for the mean density in question. 

(776 g.) But it is also possible to descend below, as well 
as to rise above, the general sea-level, and to observe the 
pendulum at great depths below that level, as in deep 
mines. It was shown by Newton 16 that the attraction of 
a hollow, spherical, homogeneous shell on a point however 
situate within it, is simply nil, i.e. that the material point 
so placed is equally attracted by it in all directions. Hence 
by descending below the surface, we set ourselves free of 
the attraction of the whole spherical shell exterior to the 
point of observation, and the remaining attraction is the 
same as that of the whole interior mass collected in its 
centre. This may, or may not, be less than the attraction 
of the whole earth on a point at its surface. It will be less 
if the earth be homogeneous or of the same density through- 
out; for in that case Newton has shown 17 that the attractive 
forces of the whole sphere, and of the interior sphere, each 
on a point on its own surface, are to each other as their 
radii. But if the internal portions of the earth be more 
dense than the external (as they must be if the foregoing 
determinations be any approach to truth), it may be greater. 
The experiment has been made, on three several occasions, 
by the present Astronomer Koyal (Mr. Airy). On the first 
two in the Dolcoath mine in Cornwall at a depth of 1200 
feet — a clock and pendulum were transported alternately 
to the bottom and the mouth of the shaft. On both these 
occasions the arrangements were defeated; on the first, by 
the accidental combustion of the packages of instruments 
in mid-air while in the act of raising them from below, 

16 Princip. Lib. i. Prop. 70. « lb., 73. 



OUTLINES OF ASTRONOMY 687 

attended with their precipitation down the shaft of the 
mine; on the other, by the subsidence of a mass of rock, 
"many times the size of Westminster Abbey," during the 
experiments, deluging the mine with water and forcing a 
premature conclusion. The third attempt (in the Harton 
Coal Pit, near South Shields, 1200 feet in depth) proved 
perfectly successful, and the oscillations of the pendulum 
below being compared with those of the clock above, by 
the immediate transmission of the beats of the latter down 
the mine by an electric wire, the great difficulty (that of 
the exact transmission of time) was annihilated. The result 
of this experiment was that a pendulum vibrating seconds 
at the mouth of the pit, would gain 2\ sec. per day at its 
bottom; and the final result (of which the calculations have 
very recently been published 18 ) gives 6*565 for the mean 
density of the earth. 

(776 h.) The difference between these several results is 
considerable, and even the interval between the last men- 
tioned and the highest of the others pretty large: it is 
bridged over, however, so to speak, and the interval partly 
filled up, by the results of a totally different class of ex- 
periments of a much more curious and artificial nature, 
which we have now to describe. We have already seen 
(art. 234) that the force of gravity may be brought directly 
into comparison with other material forces by using as an 
intermedium the elastic power of a spring. What is true 
of gravitation to the whole mass of the earth is equally so 
of gravitation toward any material mass, as a leaden ball. 
It may be measured by equilibrating it with the tension of 
a spring; provided, 1st, that we can frame a spring so deli- 

18 Phil. Trans. 1856, p. 297. 



688 



OUTLINES OF ASTRONOMY 



cate as to be visibly and measurably affected by so minute 
a force; 2dly, that the force can be so applied as to be the 
only force tending to bend the spring, a condition which 
implies that it shall act on it, not vertically, but horizon- 
tally, so as to eliminate the weight of the spring, or at least 
prevent its being mixed up with the result; and, 3dly, that 
we shall possess some independent means of measuring the 
elastic power of the spring itself. All these conditions are 
satisfied by the balance of torsion, devised by Michell with 
a view to this inquiry, and applied, after his death, to the 
intended purpose by Cavendish, in the celebrated experi- 
ment usually cited as "the Cavendish Experiment." 19 

(776 L) The apparatus consists essentially of a long 
wooden rod made so as to unite great strength with little 
weight; carrying at its extremities two equal balls A, B, 



and suspended in a horizontal situation by a wire no thicker 
than necessary securely to sustain the weight, from a point 
over its centre of gravity, the wire being arranged as in the 
figure, so as to relieve the rod of the weight of the balls, its 



19 Phil. Trans. 1Y98, 469. Cavendish expressly states that Michell's in- 
vention of this beautiful instrument, and his communication of it to him, was 
antecedent to the publication of Coulomb's researches. 



OUTLINES OF ASTRONOMY 689 

office being solely to keep them apart at a given horizontal 
distance. It is evident that when suspended from C, and 
allowed to take its position of equilibrium undisturbed bj 
any external force, the rod will assume such a situation 
that the wire C D shall be quite devoid of torsion; but that 
if the rod A B be disturbed from this neutral position, C D 
remaining vertical, the elastic force of the wire called into 
action by the torsion so induced will tend to bring it back 
to the point of departure by a force proportional to the 
angle of torsion. When so disturbed then, and abandoned 
to itself, it will oscillate backward and forward in horizontal 
arcs, the oscillations being all performed in equal times; 
and from the time observed to be occupied in each oscilla- 
tion, the weights of the balls and that of the rod being 
known, we are able, from dynamical principles, to deter- 
mine the motive force by which the wire acts on the balls, 
or the force of torsion. Suppose, now, two heavy leaden 
spheres to be brought, laterally, up nearly into contact, the 
one with A, the other with B, but on opposite sides 01 
them, they will attract A, B, and their attractions will con- 
spire in twisting the wire the same way; so that the point 
of rest will be changed from the original neutral point to 
one in which the torsion shall just counterbalance the 
attractions. By shifting the attracting balls alternately 
to the one and the other sides of A B, these will assume 
positions of rest, alternately on opposite sides of the original 
neutral point, and equidistant from it, so that the deviation, 
if any, shall thus become doubled in its effect on the read- 
ings off of a scale marked by a pointer at the end of the 
rod, which may be observed through a telescope placed at 
a distance, so that the approach of the observer's person 
may create no disturbance. 



690 OUTLINES OF ASTRONOMY 

(776/.) Practically, the observation is not so simple as 
in the above statement. The balls can hardly ever be 
brought completely to rest; and the neutral point has to 
be concluded by noting the extremes of the arc of oscilla- 
tion, perpetually diminishing by the resistance of the air, 
And when the attracting balls are brought into action, their 
attraction (acting laterally, according to the inverse squares 
of the distances) mixes itself with the force of torsion, to 
produce a compound law of force, under whose influence 
the times, velocities, and arcs have a different relation from 
those due to the torsion alone, and which, when investigated 
rigorously, lead to calculations of great complexity. Fortu- 
nately, the extreme minuteness of the attractive forces dis- 
penses with a rigorous solution of this problem, and allows 
of a very simple and ready approximation, quite exact 
enough for the purpose. But besides these, a host of dis- 
turbing influences, arising from currents of air induced by 
difference of temperature, has to be contended with or 
guarded against, so as to render the experiment one of great 
difficulty and full of niceties, the mere enumeration of 
which here, however, would lead us far beyond our limits. 20 

(776 Jc.) The experiment, as conducted by Cavendish, 
afforded as its final result 5*480. Kepeated since, with 
greater precautions, by Professor Keich, 5438 was ob- 
tained; and still more recently, by the late F. Baily, in 
a series of experiments exhibiting an astonishing amount 
of skill and patience in overcoming the almost innumerable 



20 The reader is warned to be on his guard against accepting as correct an 
account of the principle of the Cavendish experiment, professing to emanate 
from one very high astronomical authority, and passed without note or com- 
ment (and therefore so far sanctioned) by another, but which involves a total 
misconception of its true nature (Arago, Lezione di Astronomia tradutte ed 
annotate di E. Capocci, JSTapoli, 1851, p. 238). 



OUTLINES OF ASTRONOMY 691 

obstacles to complete success, 5*660; a result undoubtedly 
preferable to the two former. Thus the final result of the 
whole inquiry will stand as below, the densities concluded 
being arranged in order of magnitude: 

Schehallien experiment, by Maskelyne, calculated by Playfair, D = 4*713 

Carlini from pendulum on Mount Cenis (corrected by Giulio) . 4*950 

Col. James from attraction of Arthur's Seat .... 5-316 

"Reich, repetition of Cavendish experiment 5-438 

Cavendish, result 5*480, corrected by Mr. Baily's recomputation . 5-448 

Bailv's repetition of Cavendish experiment .... 5-660 

Airy from pendulum in Harton coal-pit ..... 6*565 

General mean . . . 5-441 21 

Mean of greatest and least . 5*639 

(776 I.) Calculating on 5.* as a result sufficiently approxi- 
mative and convenient for memory; taking the mean diam- 
eter of the earth, considered as a sphere, at 7912-41 miles, 
and the weight of a cubic foot of water at 62-3211 lbs.; we 
find for its solid content in cubic miles, 259,373 millions, 
and for its weight in tons of 2240 lbs. avoird. each, 5842 
trillions (=5842 xlO 18 ). 



21 Newton, by one of his astonishing divinations, had already expressed his 
opinion that the mean density of the earth would be found to be between five 
and six times that of water. (Princ. hi. 10.) 



692 OUTLINES OF ASTRONOMY 



PART III 
OF SIDEREAL ASTRONOMY 

CHAPTEE XY 

Of the Fixed Stars — Their Classification by Magnitudes — Photometric Scale 
of Magnitudes — Conventional or Yulgar Scale — Photometric Compari- 
son of Stars — Distribution of Stars over the Heavens — Of the Milky- 
Way or Galaxy — Its Supposed Porm that of a Flat Stratum Partially 
Subdivided — Its Visible Course among the Constellations — Its Internal 
Structure— Its Apparently Indefinite Extent in Certain Directions — Of 
the Distance of the Fixed Stars — Their Annual Parallax — Parallactic 
Unit of Sidereal Distance — Effect of Parallax Analogous to that of 
Aberration — How Distinguished from it — Detection of Parallax by 
Meridional Observations — Henderson's Application to <*■ Centauri — 
By Differential Observations— Discoveries of Bessel and Struve — 
List of Scars iu which Parallax has been Detected — Of the Real 
Magnitudes of the Stars — Comparison of their Lights with that of 
the Sun 

(777.) Besides the bodies we have described in the fore- 
going chapters, the heavens present us with an innumerable 
multitude of other objects, which are called generally by the 
name of stars. Though comprehending individuals differ- 
ing from each other, not merely in brightness, but in many 
other essential points, they all agree in one attribute — a 
high degree of permanence as to apparent relative situation. 
This has procured them the title of "fixed stars"; an ex- 
pression which is to be understood in a comparative and not 
an absolute sense, it being certain that many, and probable 
that all, are in a state of motion, although too slow to be 
perceptible unless by means of very delicate observations, 
continued during a long series of years. 

(778.) Astronomers are in the habit of distinguishing the 



OUTLINES OF ASTRONOMY 693 

stars into classes, according to their apparent brightness. 
These are termed magnitudes. The brightest stars are said 
to be of the first magnitude; those which fall so far short of 
the first degree of brightness as to make a strongly marked 
distinction are classed in the second; and so on down to the 
sixth or seventh, which comprise the smallest stars visible 
to the naked eye, in the clearest and darkest night. Beyond 
these, however, telescopes continue the range of visibility, 
and magnitudes from the 8th down to the 16th are familiar 
to those who are in the practice of using powerful instru- 
ments; nor does there seem the least reason to assign a limit 
to this progression; every increase in the dimensions and 
power of instruments, which successive improvements in 
optical science have attained, having brought into view 
multitudes innumerable of objects invisible before; so that, 
for anything experience has hitherto taught us, the number 
of the stars may be really infinite, in the only sense in which 
we can assign a meaning to the word. 

(779.) This classification into magnitudes, however, it 
must be observed, is entirely arbitrary. Of a multitude of 
bright objects, differing probably, intrinsically, both in size 
and in splendor, and arranged at unequal distances from us, 
one must of necessity appear the brightest, one next below 
it, and so on. An order of succession (relative, of course, 
to our local situation among them) must exist, and it is a 
matter of absolute indifference, where, in that infinite pro- 
gression downward, from the one brightest to the invisible, 
we choose to draw our lines of demarcation. All this is 
a matter of pure convention. Usage, however, has estab- 
lished such a convention; and though it is impossible to 
determine exactly, or d priori, where one magnitude ends 
and the next begins, and although different observers have 



694 OUTLINES OF ASTRONOMY 

differed in their magnitudes, yet, on the whole, astronomers 
have restricted their first magnitude to about 23 or 24 prin- 
cipal stars; their second to 50 or 60 next inferior; their 
third to about 200 jet smaller, and so on; the numbers in- 
creasing very rapidly as we descend in the scale of bright- 
ness, the whole number of stars already registered down to 
the seventh magnitude, inclusive, amounting to from 12000 
to 15000. 

(780.) As we do not see the actual disk of a star, but 
judge only of its brightness by the total impression made 
upon the eye, the apparent "magnitude" of any star will, it 
is evident, depend, 1st, on the star's distance from us; 2d, 
on the absolute magnitude of its illuminated surface; 3d, on 
the intrinsic brightness of that surface. Now, as we know 
nothing, or next to nothing, of any of these data, and have 
every reason for believing that each of them may differ in 
different individuals, in the proportion of many millions to 
one, it is clear that we are not to expect much satisfaction 
in any conclusions we may draw from numerical statements 
of the number of individuals which have been arranged in 
our artificial classes antecedent to any general or definite 
principle of arrangement. In fact, astronomers have not 
yet agreed upon any principle by which the magnitudes 
may be photometrically classed d priori, whether for ex- 
ample a scale of brightnesses decreasing in geometrical pro- 
gression should be adopted, each term being one-half of the 
preceding, or one- third, or any other ratio, or whether it 
would not be preferable to adopt a scale decreasing as the 
squares of the terms of a harmonic progression, i.e. accord- 
ing to the series 1, J, J, i, M , etc. The former would be a 
purely photometric scale, and would have the apparent ad- 
vantage that the light of a star of any magnitude would bear 



OUTLINES OF ASTRONOMY 

a fixed proportion to that of the magnitude next above it, 
an advantage, however, merely apparent, as it is certain, 
from many optical facts, that the unaided eye forms very 
different judgments of the proportions existing between 
bright lights, and those between feeble ones. The latter 
scale involves a physical idea, that of supposing the scale 
of magnitudes to correspond to the appearance of a first 
magnitude standard star, removed successively to twice, 
three times, etc., its original or standard distance. Such a 
scale, which would make the nominal magnitude a sort of 
index to the presumable or average distance, on the supposi- 
tion of an equality among the real lights of the stars, would 
facilitate the expression of speculative ideas on the constitu- 
tion of the sidereal heavens. On the other hand, it would 
at first sight appear to make too small a difference between 
the lights in the lower magnitudes. For example, on this 
principle of nomenclature, the light of a star of the seventh 
magnitude would be thirty-six 49ths of that of one of the 
sixth, and of the tenth 81 hundredths of the ninth, while be- 
tween the first and the second the proportion would be that 
of four to one. So far, however, from this being really ob- 
jectionable, it falls in well with the general tenor of the 
optical facts already alluded to, inasmuch as the eye (in 
the absence of disturbing causes) does actually discriminate 
with greater precision between the relative intensities of 
feeble lights than of bright ones, so that the fraction If, for 
instance, expresses quite as great a step downward (physio- 
logically speaking) from the sixth magnitude, as J does from 
the first. As the choice, therefore, so far as we can see, 
lies between these two scales, in drawing the lines of demar- 
cation between what may be termed the photornetrical mag- 
nitudes of the stars, we have no hesitation in adopting, and 



698 OUTLINES OF ASTRONOMY 

recommending others to adopt, the latter system in prefer- 
ence to the former. 

(781.) The conventional magnitudes actually in use 
among astronomers, so far as their usage is consistent with 
itself, conforms moreover very much more nearly to this 
than to the geometrical progression. It has been shown 1 
by direct photometric measurement of the light of a consid- 
erable number of stars from the first to the fourth magni- 
tude, that if M be the number expressing the magnitude of 
a star on the above system, and m the number expressing 
the magnitude of the same star in the loose and irregular 
language at present conventionally or rather provisionally 
adopted, so far as it can be collected from the conflicting 
authorities of different observers, the difference between 
these numbers, or M — m, is the same in all the higher parts 
of the scale, and is less than half a magnitude (0 m - 414). 
The standard star assumed as the unit of magnitude in the 
measurements referred to, is the bright southern star a Cen- 
tauri, a star somewhat superior to Arcturus in lustre. If 
we take the distance of this star for unity, it follows that 
when removed to the distances 1*414, 2-414, 3*414, etc., its 
apparent lustre would equal those of average stars of the 
1st, 2d, 3d, etc., magnitudes, as ordinarily reckoned, 
respectively. 

(782.) The difference of lustre between stars of two con- 
secutive magnitudes is so considerable as to allow of many 
intermediate gradations being perfectly well distinguished. 
Hardly any two stars of the first or of the second magnitude 
would be judged by an eye practiced in such comparisons 



1 See "Results of Observations made at the Cape of Good Hope," etc., 
p. 371. By the Author. 



OUTLINES OF ASTRONOMY 697 

to be exactly equal in brightness. Hence, the necessity, if 
anything like accuracy be aimed at, of subdividing the mag- 
nitudes and admitting fractions into our nomenclature of 
brightness. When this necessity first began to be felt, a 
simple bisection of the interval was recognized, and the 
intermediate degree of brightness was thus designated, viz. 
1.2 m, 2.3 m, and so on. At present it is not infrequent to 
find the interval trisected thus: 1 m, 1.2 m, 2.1 m, 2 m, etc., 
where the expression 1.2 m denotes a magnitude intermedi- 
ate between the first and second, but nearer 1 than 2; while 
2.1 m designates a magnitude also intermediate, but nearer 
2 than 1. This may suffice for common parlance, but as 
this department of astronomy progresses toward exactness, 
a decimal subdivision will of necessity supersede these rude 
forms of expression, and the magnitude will be expressed 
by an integer number followed by a decimal fraction; as, 
for instance, 2.51, which expresses the magnitude of y Gremi- 
norum on the vulgar or conventional scale of magnitudes, 
by which we at once perceive that its place is almost ex- 
actly half way between the 2d and 3d average magnitudes, 
and that its light is to that of an average first magnitude 
star in that scale (of which a Orionis in its usual or normal 
state 2 may be taken as a typical specimen) as l 2 : (2*51) 2 , and 
to that of a Centauri as l 2 : (2'924) 2 , making its place in the 
photometric scale (so defined) 2-924. Lists of stars northern 
and southern, comprehending those of the vulgar first, sec- 
ond, and third magnitudes, with their magnitudes decimally 
expressed in both systems, will be found at the end of this 
work. The light of a star of the sixth magnitude may be 
roughly stated as about the hundredth part of one of the 

2 In the interval from 1836 to 1839 this star underwent considerable and 
remarkable fluctuations of brightness. 



OUTLINES OF ASTRONOMY 

first. Sirius would make between three and four hundred 
stars of that magnitude. 

(783.) The exact photometrical determination of the com- 
parative intensities of light of the stars is attended with 
many and great difficulties, arising partly from their differ- 
ences of color; partly from the circumstance that no invari- 
able standard of artificial light has yet been discovered; 
partly from the physiological cause above alluded to, by 
which the eye is incapacitated from judging correctly of the 
proportion of two lights, and can only decide (and that with 
not very great precision) as to their equality or inequality; 
and partly from other physiological causes. The least ob- 
jectionable method hitherto proposed would appear to be 
the following. A natural standard of comparison is in the 
first instance selected, brighter than any of the stars, so as 
to allow of being equalized with any of them by a reduction 
of its light optically effected, and at the same time either 
invariable, or at least only so variable that its changes can 
be exactly calculated and reduced to numerical estimation. 
Such a standard is offered by the planet Jupiter, which, 
being much brighter than any star, subject to no phases, and 
variable in light only by the variation of its distance from 
the sun, and which moreover comes in succession above the 
horizon at a convenient altitude simultaneously with all 
the fixed stars, and in the absence of the moon, twilight, 
and other disturbing causes (which fatally affect all obser- 
vations of this nature), combines all the requisite conditions, 
Let us suppose, now, that Jupiter being at A and the star 
to be compared with it at B, a glass prism C is so placed 
that the light of the planet deflected by total internal reflec- 
tion at its base, shall emerge parallel to B E, the direction of 
the star's visual ray. After reflection, let it be received on 



OUTLINES OF ASTRONOMY 

a lens D, in whose focus F it will form a small bright star- 
hke image capable of being viewed by an eye placed at E, 
so far out of the axis of the cone of diverging rays as to 
admit of seeing at the same time, and with the same eye, 
and so comparing, this image with the star seen directly. 




By bringing the eye nearer to or further from the focus F, 
the apparent brightness of the focal point will be varied in 
the inverse ratio of the square of the distance E F, and 
therefore may be equalized, as well as the eye can judge of 
such equalities, with the star. If this be done for two stars 
several times alternately, and a mean of the results taken, 
by measuring E F, their relative brightness will be obtained: 
that of Jupiter, the temporary standard of comparison, being 
altogether eliminated from the result. 

(784.) A moderate number of well-selected stars being 
thus photometrically determined by repeated and careful 
measurements, so as to afford an ascertained and graduated 
scale of brightness among the stars themselves, the inter- 
mediate steps or grades of magnitude may be filled up, by 
inserting between them, according to the judgment of the 
eye, other stars, forming an ascending or descending 
sequence, each member of such a sequence being brighter 
than that below, and less bright than that above it; and 
thus at length, a scale of numerical magnitudes will be- 

ASTKONOMT— Vol. XX— 11 



700 OUTLINES OF ASTRONOMY 

come established, complete in all its members, from Sirius, 
the brightest of the stars, down to the least visible magni- 
tude. 3 It were much to be wished that this branch of as- 
tronomy, which at present can hardly be said to be advanced 
beyond its infancy, were perseveringly and systematically 
cultivated. It is by no means a subject of mere barren 
curiosity, as will abundantly appear when we come to 
speak of the phenomena of variable stars; and being more- 
over one in which amateurs of the science may easily chalk 
out for themselves a useful and available path, may natu- 
rally be expected to receive large and interesting accessions 
at their hands. 

(785.) If the comparison of the apparent magnitudes of 
the stars with their numbers leads to no immediately obvious 
conclusion, it is otherwise when we view them in connection 
with their local distribution over the heavens. If indeed 
we confine ourselves to the three or four brightest classes, 
we shall find them distributed with a considerable approach 
to impartiality over the sphere: a marked preference how- 
ever being observable, especially in the southern hemi- 
sphere, to a zone or belt, following the direction of a great 
circle passing through e Orionis and a Crucis. But if we 
take in the whole amount visible to the naked eye, we shall 
perceive a great increase of number as we approach the 
borders of the Milky Way. And when we come to tele- 
scopic magnitudes, we find them crowded beyond imagina- 

3 For the method of combining and treating such sequences, where accumu- 
lated in considerable numbers, so as to eliminate from their results the influence 
of erroneous judgment, atmospheric circumstances, etc., which often give rise to 
contradictory arrangements in the order of stars differing but little in magnitude, 
as well as for an account of a series of photometric comparisons (in which, how- 
ever, not Jupiter, but the moon, was used as an intermediate standard), see the 
work above cited, note on p. 353. (Results of Observations, etc.) Prof. Heis 
of Munster is, so far as we are aware, the only observer who has adopted and 
extended the method of sequences there employed. 



OUTLINES OF ASTRONOMY 70l 

tion, along the extent of that circle, and of the branches 
which it sends off from it; so that in fact its whole light ig 
composed of nothing but stars of every magnitude, from 
such as are visible to the naked eye down to the smallest 
point of light perceptible with the best telescopes. 

(786.) These phenomena agree with the supposition that 
the stars of our firmament, instead of being scattered in 
all directions indifferently through space, form a stratum 
of which the thickness is small, in comparison with its 
length and breadth; and in which the earth occupies a 
place somewhere about the middle of its thickness, and 
near the point where it subdivides into two principal 
laminae, inclined at a small angle to each other (art. 802). 
For it i3 certain that, to an eye so situated, the apparent 




density of the stars, supposing them pretty equally scat= 
tered through the space they occupy, would be least in a 
direction of the visual ray (as S A), perpendicular to the 
lamina, and greatest in that of its breadth, as S B, S C, SD; 
increasing rapidly in passing from one to the other direction, 
just as we see a slight haze in the atmosphere thickening 
into a decided fog bank near the horizon, by the rapid in- 
crease of the mere length of the visual ray. Such is the 
view of the construction of the starry firmanent taken by 
Sir William Herschel, whose powerful telescopes first 
effected a complete analysis of this wonderful zone, and 
demonstrated the fact of its entirely consisting of stars. 4 

4 Thomas Wright of Durham (Theory of the Universe, London, 1*750) ap- 
pears so early as 1734 to have entertained the same general view as to the con- 



702 OUTLINES OF ASTRONOMY 

So crowded are they in some parts of it, that by counting 
the stars in a single field of his telescope, he was led to 
conclude that 50000 had passed under his review in a zone 
two degrees in breadth, during a single hour's observation. 
In that part of the Milky Way which is situated in lOh. 30m. 
E. A. and between the 147th and 150th degree of N. P. D., up- 
ward of 5000 stars have been reckoned to exist in a square 
degree. The immense distances at which the remoter re- 
gions must be situated will sufficiently account for the vast 
predominance of small magnitudes which are observed in it. 
(787.) The course of the Milky Way as traced through 
the heavens by the unaided eye, neglecting occasional 
deviations and following the line of its greatest brightness 
as well as its varying breadth and intensity will permit, 
conforms as nearly as the indefiniteness of its boundary will 
allow it to be fixed, to that of a great circle inclined at an 
angle of about 63° to the equinoctial, and cutting that circle 
in E. A. 6h. 47m. and 18h. 47m., so that its northern and 
southern poles respectively are situated in E. A. 12h. 47m. 
K. P. D. 63° and E. A. Oh. 47m. N. P. D. 117°. Throughout 
the region where it is so remarkably subdivided (art. 186), 
this great circle holds an intermediate situation between the 
two great streams; with a nearer approximation however to 
the brighter and continuous stream, than to the fainter and 
interrupted one. If we trace its course in order of right 
ascension, we find it traversing the constellation Cassiopeia, 
its brightest part passing about two degrees to the north of 
the star 3 of that constellation, i.e. in about 62° of north 



stitution of the Milky Way and starry firmament, founded, quite in the spirit of 
just astronomical speculation, on a partial resolution of a portion of it with a 
" one-foot reflector" (a reflector one foot in focal length). See an account of this 
rare work by Mr. de Morgan in Phil. Mag. Ser. 3, xxxii. p. 241, et seq. 



OUTLINES OF ASTRONOMY 703 

declination, or 28° N. P. D. Passing thence between y and 
e Cassiopeia? it sends off a branch to the south-preceding 
side, toward a Persei, very conspicuous as far as that star, 
prolonged faintly toward e of the same constellation, and 
possibly traceable toward the Hyades and Pleiades as re- 
mote outliers. The main stream, however (which is here 
very faint), passes on through Auriga, over the three re- 
markable stars, e, C, h, of that constellation called the Hcedi, 
preceding Capella, between the feet of Gemini and the horns 
of the Bull (where it intersects the ecliptic nearly in the 
Solstitial Colure) and thence over the club of Orion to 
the neck of Monoceros, intersecting the equinoctial in 
E. A. 6h. 54m. Up to this point, from the offset in Per- 
seus, its light is feeble and indefinite, but thenceforward 
it receives a gradual accession of brightness, and where it 
passes through the shoulder of Monoceros and over the 
head of Canis Major it presents a broad, moderately bright, 
very uniform, and to the naked eye, starless stream up to 
the point where it enters the prow of the ship Argo, 
nearly on the southern tropic. 5 Here it again subdivides 
(about the star m Puppis), sending off a narrow and winding 
branch on the preceding side as far as y Argus, where it 
terminates abruptly. The main stream pursues its south- 
ward course to the 123d parallel of N.P.D., where it diffuses 
itself broadly and again subdivides, opening out into a wide 
fan-like expanse, nearly 20° in breadth formed of interlacing 
branches, all which terminate abruptly, in a line drawn 
nearly through k and y Argus. 

5 In reading this description a celestial globe will be a necessary companion. 
It may be thought needless to detail the course of the Milky Way verbally, 
since it is mapped down on all celestial charts and globes. But in the gen- 
erality of them, indeed in all which have come to our knowledge, this is done 
so very loosely and incorrectly, as by no means to dispense with a verbal 
description. 



704 OUTLINES OF ASTRONOMY 

(788.) At this place the continuity of the Milky Way is 
interrupted by a wide gap, and where it recommences on 
the opposite side it is by a somewhat similar fan-shaped 
assemblage of branches which converge upon the bright 
star 7) Argus. Thence it crosses the hindfeet of the Cen- 
taur, forming a curious and sharply defined semicircular 
concavity of small radius, and enters the Cross by a very 
bright neck or isthmus of not more than 3 or 4 degrees in 
breadth, being the narrowest portion of the Milky Way. 
After this it immediately expands into a broad and bright 
mass, inclosing the stars a and p Crucis, and p Centauri, and 
extending almost up to a of the latter constellation. In the 
midst of this bright mass, surrounded by it on all sides, 
and occupying about half its breadth, occurs a singular 
dark pear-shaped vacancy, so conspicuous and remarkable 
as to attract the notice of the most superficial gazer, and 
to have acquired among the early southern navigators the 
uncouth but expressive appellation of the coal-sack. In this 
vacancy which is about 8° in length, and 5° broad, only one 
very small star visible to the naked eye occurs, though it 
is far from devoid of telescopic stars, so that its striking 
blackness is simply due to the effect of contrast with the 
brilliant ground with which it is on all sides surrounded. 
This is the place of nearest approach of the Milky Way to 
the South Pole. Throughout all this region its brightness 
is very striking, and when compared with that of its more 
northern course already traced, conveys strongly the im- 
pression of greater proximity, and would almost lead to 
a belief that our situation as spectators is separated on 
all sides by a considerable interval from the dense body 
of stars composing the Galaxy, which in this view of the 
subject would come to be considered as a flat ring or some 



OUTLINES OF ASTRONOMY 705 

other re-entering form of immense and irregular breadth 
and thickness, within which we are excentrically situated, 
nearer to the southern than to the northern part of its 
circuit. 

(789.) At a Centauri, the Milky Way again subdivides, 6 
sending off a great branch of nearly half its breadth, but 
which thins off rapidly, at an angle of about 20° with its 
general direction, toward the preceding side, to ^ and d 
Lupi, beyond which it loses itself in a narrow and faint 
streamlet. The main stream passes on increasing in breadth 
to y Normse, where it makes an abrupt elbow and again sub- 
divides into one principal and continuous stream of very 
irregular breadth and brightness on the following side, and 
a complicated system of interlaced streaks and masses on 
the preceding, which covers the tail of Scorpio, and termi- 
nates in a vast and faint effusion over the whole extensive 
region occupied by the preceding leg of Ophiuchus, extend- 
ing northward to the parallel of 103° N". P. D., beyond which 
it cannot be traced; a wide interval of 14°, free from all ap- 
pearance of nebulous light, separating it from the great 
branch on the north side of the equinoctial of which it is 
usually represented as a continuation. 

(790.) Eeturning to the point of separation of this great 
branch from the main stream, let us now pursue the course 
of the latter. Making an abrupt bend to the following side, 
it passes over the stars t Arse, and i Scorpii, and y Tubi to 
Y Sagittarii, where it suddenly collects into a vivid oval 
mass about 6° in length and 4° in breadth, so excessively 
rich in stars that a very moderate calculation makes their 
number exceed 100,000. Northward of this mass, this 

6 All the maps and globes place this subdivision at £ Centauri, but erro- 
neously. 



706 OUTLINES OF ASTRONOMY 

stream crosses the ecliptic in longitude about 276°, and 
proceeding along the bow of Sagittarius into Antinous has 
its course rippled by three deep concavities, separated from 
each other by remarkable protuberances, of which the larger 
and brighter (situated between Flamsteed's stars 3 and 6 
Aquilse) forms the most conspicuous patch in the southern 
portion of the Milky Way visible in our latitudes. 

(791.) Crossing the equinoctial at the 19th hour of right 
ascension, it next runs in an irregular, patchy, and winding 
stream through Aquila, Sagitta and Vulpecula up to Cyg- 
nus; at e of which constellation its continuity is interrupted, 
and a very confused and irregular region commences, 
marked by a broad dark vacuity, not unlike the southern 
"coal-sack," occupying the space between e, a, and y Cygni, 
which serves as a kind of centre for the divergence of three 
great streams; one, which we have already traced; a second, 
the continuation of the first (across the interval) from a 
northward, between Lacerta and the head of Cepheus to the 
point in Cassiopeia whence we set out, and a third branch- 
ing off from y Cygni, very vivid and conspicuous, running 
off in a southern direction through p Cygni, and s Aquilae 
almost to the equinoctial, where it loses itself in a region 
thinly sprinkled with stars, where in some maps the modern 
constellation Taurus Poniatovski is placed. This is the 
branch which, if continued across the equinoctial, might 
be supposed to unite with the great southern effusion in 
Ophiuchus already noticed (art. 789). A considerable off- 
set, or protuberant appendage, is also thrown off by the 
northern stream from the head of Cepheus directly toward 
the pole, occupying the greater part of the quartile formed 
by a, p, i, and d of that constellation. 

(792.) We have been somewhat circumstantial in de- 



OUTLINES OF ASTRONOMY 707 

scribing the course and principal features of the Via Lac- 
tea, not only because there does not occur anywhere (so far 
as we know) any correct account of it, but chiefly by reason 
of its high interest in sidereal astronomy, and that the reader 
may perceive how very difficult it must necessarily be to 
form any just conception of the real, solid form, as it exists 
in space, of an object so complicated, and which we see from 
a point of view so unfavorable. The difficulty is of the 
same kind which we experience when we set ourselves to 
conceive the real shape of an auroral arch or of the clouds, 
but far greater in degree, because we know the laws which 
regulate the -formation of the latter, and limit them to cer- 
tain conditions of altitude — because their motion presents 
them to us in various aspects, but chiefly because we con- 
template them from a station considerably below their gen- 
eral plane, so as to allow of our mapping out some kind of 
ground-plan ol their shape. All these aids are wanting 
when we attempt to map and model out the Gralaxy, and 
beyond the obvious conclusion that its form must be, gen- 
erally speaking, flat, and of a thickness small in comparison 
with its area in length and breadth, the laws of perspective 
afford us little further assistance in the inquiry. Probabil- 
ity may, it is true, here and there enlighten us as to certain 
features. Thus when we see, as in the coal-sack, a sharply 
denned oval space free from stars, insulated in the midst of 
a uniform band of not much more than twice its breadth, it 
would seem much less probable that a conical or tubular 
hollow traverses the whole of a starry stratum, continuously 
extended from the eye outward, than that a distant mass of 
comparatively moderate thickness should be simply perfo- 
rated from side to side, or that an oval vacuity should be 
seen foreshortened, in a distant foreshortened area, not really 



708 OUTLINES OF ASTRONOMY 

exceeding two or three times its own breadth. Neither can 
we without obvious improbability refuse to admit that the 
long lateral offsets which at so many places quit the main 
stream and run out to great distances, are either planes seen 
edgewise, or the convexities of curved surfaces viewed tan- 
gentially, rather than cylindrical or columnar excrescences 
bristling up obliquely from the general level. And in the 
same spirit of probable surmise we may account for the in- 
tricate reticulations above described as existing in the region 
of Scorpio, rather by the accidental crossing of streaks thus 
originating, at very different distances, or by a cellular 
structure of the mass, than by real intersections. Those 
cirrous clouds which are often seen in windy weather, con- 
vey no inapt impression either of the kind of appearance in 
question, or of the structure it suggests. It is to other indi- 
cations, however, and chiefly to the telescopic examination 
of its intimate constitution and to the law of the distribution 
of stars, not only within its bosom, but generally over the 
heavens, that we must look for more definite knowledge 
respecting its true form and extent. 

(793.) It is on observations of this latter class, and not on 
merely speculative or conjectural views, that the generali- 
zation in art. 786, which refers the phenomena of the starry 
firmament to the system of the Galaxy as their embodying 
fact, is brought to depend. The process of "gauging" the 
heavens was devised by Sir W. Herschel for this purpose. 
It consisted in simply counting the stars of all magnitudes 
which occur in single fields of view, of 15' in diameter, visi- 
ble through a reflecting telescope of 18 inches aperture, and 
20 feet focal length, with a magnifying power of 180° : the 
points of observation being very numerous and taken indis- 
criminately in every part of the surface of the sphere visible 



OUTLINES OF ASTRONOMY 70^ 

in our latitudes. On a comparison of many hundred such 
"gauges" or local enumerations it appears that the density 
of starlight (or the number of stars existing on an average 
of several such enumerations in any one immediate neigh- 
borhood) is least in the pole of the Galactic circle, 1 and in- 
creases on all sides, with the Galactic polar distance (and 
that nearly equally in all directions) down to the Milky 
Way itself, where it attains its maximum. The progressive 
rate of increase in proceeding from the pole is at first slow, 
but becomes more and more rapid as we approach the plane 
of that circle according to a law of which the following num- 
bers, deduced by M. Struve from a careful analysis of all 
the gauges in question, will afford a correct idea: 

o.i.»«t;,. 8 vr„„ t u t>,o q „ r« a + Q , 1 „ Q Average Number of Stars in a 

Galactic North Polar Distance F ? eld 15 , in Diameter 

0° 4-15 

15° 4-68 

30° 6-52 

45° 10 36 

60° 1*7-68 

75° 30-30 

90° 122-00 

From which it appears that the mean density of the stars in 
the galactic circle exceeds in a ratio of very nearly 30 to 1 
that in its pole, and in a proportion of more than 4 to 1 that 
in a direction 15° inclined to its plane. 

(794.) These numbers fully bear out the statement in 
art. 786 and even draw closer the resemblance by which 
that statement is there illustrated. For the rapidly increas- 
ing density of a fog-bank as the visual ray is depressed 
toward the plane of the horizon is a consequence not only 

7 From ya\a, ya\a.KTo<;, milk; meaning the great circle spoken of in art. 787, 
to which the course of the Via Lactea most nearly conforms. This circle is to 
sidereal what the invariable ecliptic is to planetary astronomy — a plane of ulti- 
mate reference, the ground -plane of the sidereal system. 

8 Etudes d'Astronomie Stellaire, p. 71. M. Struve maintains the Galactic 
circle to be a small, not a great, circle of the sphere. The appeal is to the eye- 
sight. I retain my own conviction. 



710 



OUTLINES OF ASTRONOMY 



of the mere increase in length of the foggy space traversed, 
but also of an actual increase of density in the fog itself in 
its lower strata. Now this very conclusion follows from a 
comparison inter se of the numbers above set down, as M. 
Struve has clearly shown from a mathematical analysis of 
the empirical formula, which faithfully represents their law 
of progression, and of which he states the result in the fol- 
lowing table, expressing the densities of the stars at the re- 
spective distances, 1, 2, 3, etc., from the galactic plane, taking 
the mean density of the stars in that plane itself for unity. 



Distances from the 


Density of 


Distances from the 


Density of 


Galactic Plane. 


Stars. 


Galactic Plane. 


Stars. 


o-oo 


1-00000 


0-50 


0-08646 


0-05 


0-48568 


0-60 


0-05510 


o-io 


0-33288 


0-70 


0-03079 


0-20 


0-23895 


0-80 


0-01414 


0-30 


0-17980 


0-866 


0.00532 


0-40 


013021 







The unit of distance, of which the first column of this table 
expresses fractional parts, is the distance at which such a 
telescope is capable of rendering just visible a star of average 
magnitude, or, as it is termed, its space-penetrating power. 
As we ascend therefore from the galactic plane into this 
kind of stellar atmosphere, we perceive that the density of 
its parallel strata decreases with great rapidity. At an alti- 
tude above that plane equal to only one-twentieth of the 
telescopic limit, it has already diminished to one-half, and 
at an altitude of 0-866, to hardly more than one-two-hun- 
dredth of its amount in that plane. So far as we can per- 
ceive there is no flaw in this reasoning, if only it be granted, 
1st, that the level planes are continuous, and of equal den- 
sity throughout; and, 2dly, that an absolute and definite limit 



OUTLINES OF ASTRONOMY 711 

is set to telescopic vision, beyond which, if stars exist, they elude 
our sight, and are to us as if they existed not: a postulate 
whose probability the reader will be in a better condition 
to estimate, when in possession of some other particulars 
respecting the constitution of the Galaxy to be described 
presently. 

(795.) A similar course of observation followed out in 
the southern hemisphere, leads independently to the same 
conclusion as to the law of the visible distribution of stars 
over the southern galactic hemisphere, or that half of the 
celestial surface which has the south galactic pole for its 
centre. A system of gauges, extending over the whole sur- 
face of that hemisphere taken with the same telescope, field 
of view and magnifying power employed in Sir William 
Herschel's gauges, has afforded the average numbers of 
stars per field of 15' in diameter, within the areas of zones 
encircling that pole at intervals of 15°, set down in the fol- 
lowing table: 

Average Number of Stars 
per Field of 15' 

6 05 

662 

9-08 

13 49 

26-29 

59-06 

(796.) These numbers are not directly comparable with 
those of M. Struve, given in art. 793, because the latter cor- 
respond to the limiting polar distances, while these are the 
averages for the included zones. That eminent astronomer, 
however, has given a table of the average gauges appropri- 
ate to each degree of north galactic polar distance, 9 from 
which it is easy to calculate averages for the whole extent 

9 Etudes d'Astronomie Stellaire, p. 34. 



;s of Galactic South 


Polar Distance 


0° 


to 15° 


15 


to 30 


30 


to 45 


45 


to 60 


60 


to 75 


75 


to 90 



Zones of Galactic North 


Polar Distance 


0° 


to 15° 


15 


to 30 


30 


to 45 


45 


to 60 


60 


to 75 


15 


to 90 



712 OUTLINES OF ASTRONOMY 

of each zone. How near a parallel the results of this calcu- 
lation for the northern hemisphere exhibit with those above 
stated for the southern will be seen by the following table: 

Average Number of Stars 

per Field of 15' from 

M. Struve's Table 

4-32 

5-42 

8-21 

13-61 

24-09 

53-43 

It would appear from this that, with an almost exactly 
similar law of apparent density in the two hemispheres, the 
southern were somewhat richer in stars than the northern, 
which may, and not improbably does arise, from our situa- 
tion not being precisely in the middle of its thickness, but 
somewhat nearer to its northern surface. 

(797.) When examined with powerful telescopes, the con- 
stitution of this wonderful zone is found to be no less vari- 
ous than its aspect to the naked eye is irregular. In some 
regions the stars of which it is wholly composed are scat- 
tered with remarkable uniformity over immense tracts, while 
in others the irregularity of their distribution is quite as 
striking, exhibiting a rapid succession of closely clustering 
rich patches separated by comparatively poor intervals, and 
indeed in some instances by spaces absolutely dark and 
completely void of any star, even of the smallest telescopic 
magnitude. In some places not more than 40 or 50 stars on 
an average occur in a "gauge" field of 15', while in others 
a similar average gives a result of 400 or 500. Nor is less 
variety observable in the character of its different regions 
in respect of the magnitudes of the stars they exhibit, and 
the proportional numbers of the larger and smaller magni- 
tudes associated together, than in respect of their aggregate 



OUTLINES OF ASTRONOMY 713 

numbers. In some, for instance, extremely minute stars, 
though never altogether wanting, occur in numbers so 
moderate as to lead us irresistibly to the conclusion that 
in these regions we see fairly through the starry stratum, 
since it is impossible otherwise (supposing their light not 
intercepted) that the numbers of the smaller magnitudes 
should not go on continually increasing ad infinitum. In 
such cases moreover the ground of the heavens, as seen 
between the stars, is for the most part perfectly dark, which 
again would not be the case, if innumerable multitudes of 
stars, too minute to be individually discernible, existed 
beyond. In other regions we are presented with the phe- 
nomenon of an almost uniform degree of brightness of the 
individual stars, accompanied with a very even distribution 
of them over the ground of the heavens, both the larger 
and smaller magnitudes being strikingly deficient. In such 
cases it is equally impossible not to perceive that we are 
looking through a sheet of stars nearly of a size, and of no 
great thickness compared with the distance which separates 
them from us. Were it otherwise we should be driven to 
suppose the more distant stars uniformly the larger, so as 
to compensate by their greater intrinsic brightness for their 
greater distance, a supposition contrary to all probability. 
In others again, and that not infrequently, we are presented 
with a double phenomenon of the same kind, viz. a tissue 
as it were of large stars spread over another of very small 
ones, the intermediate magnitudes being wanting. The con- 
clusion here seems equally evident that in such cases we look 
through two sidereal sheets separated by a starless interval. 
(798.) Throughout by far the larger portion of the extent 
of the Milky Way in both hemispheres, the general black- 
ness of the ground of the heavens on which its stars are 



714 OUTLINES OF ASTRONOMY 

projected, and the absence of that innumerable multitude 
and excessive crowding of the smallest visible magnitudes, 
and of glare produced by the aggregate light of multitudes 
too small to affect the eye singly, which the contrary sup- 
position would appear to necessitate, must, we think, be 
considered unequivocal indications that its dimensions in 
directions where these conditions obtain, are not only not 
infinite, but that the space-penetrating power of our tele- 
scopes suffices fairly to pierce through and beyond it. It 
is but right however to warn our readers that this conclusion 
has been controverted, and that by an authority not lightly 
to be put aside, on the ground of certain views taken by 
Olbers as to a defect of perfect transparency in the celestial 
spaces, in virtue of which the light of the more distant stars 
is enfeebled more than in proportion to their distance. The 
extinction of light thus originating, proceeding in geometri- 
cal progression while the distance increases in arithmetical, 
a limit, it is argued, is placed to the space-penetrating 
powers of telescopes, far within that which distance alone 
apart from such obscuration would assign. It would lead 
us too far aside of the objects of a treatise of this nature 
to enter upon any discussion of the grounds (partly meta- 
physical) on which these views rely. It must suffice here 
to observe that the objection alluded to, if applicable to 
any, is equally so to every part of the galaxy. We are not 
at liberty to argue that at one part of its circumference 
our view is limited by this sort of cosmical veil which ex- 
tinguishes the smaller magnitudes, cuts off the nebulous 
light of distant masses, and closes our view in impenetrable 
darkness; while at another we are compelled by the clearest 
evidence telescopes can afford to believe that star-strewn 
vistas lie open, exhausting their powers and stretching out 



OUTLIXES OF ASTRONOMY 715 

beyond their utmost reach, as is proved by that very phe- 
nomenon which the existence of such a veil would render 
impossible, viz. infinite increase of number and diminution 
of magnitude, terminating in complete irresolvable nebu- 
losity. Such is, in effect, the spectacle afforded by a very 
large portion of the Milky Way in that interesting region 
near its point of bifurcation in Scorpio (arts. 789, 792) 
where, through the hollows and deep recesses of its com- 
plicated structure, we behold what has all the appearance of 
a wide and indefinitely prolonged area strewed over with 
discontinuous masses and clouds of stars which the tel- 
escope at length refuses to analyze. 10 Whatever other con- 
clusions we may draw, this must anyhow be regarded as 
the direction of the greatest linear extension of the ground 
plan of the galaxy. And it would appear to follow, also, 
as a not less obvious consequence, that in those regions 
where that zone is clearly resolved into stars well separated 
and seen projected on a blach ground, and where by conse- 
quence it is certain, if the foregoing views be correct, that 
we look out beyond them into space, the smallest visible 
stars appear as such, not by reason of excessive distance, 
but of a real inferiority of size or brightness. 31 

(799.) When we speak of the comparative remoteness of 
certain regions of the starry heavens beyond others, and 
of our own situation in them, the question immediately 

i0 It would be doing great injustice to the illustrious astronomer of Pulkova 
(whose opinion, if we here seem to controvert, it is with the utmost possible 
deference and respect) not to mention that at the time of his writing the re- 
markable essay already more than once cited, in which the views in question 
are delivered, he could not have been aware of the important facts alluded to 
in the test, the work in which they are described being then unpublished. 

11 Professor Loomis (Progress of Astronomy, 1850, p. 141), with the facts 
adduced before him, arrives at a contrary conclusion. Astronomers will judge 
of the validity of his objections. Professor Argelander (Astron. Nachr. 996) 
has cited me in support of Olbers' theory, in direct opposition to my own opin- 
ion, here (as I should have thought distinctly enough) recorded. 



716 OUTLINES OF ASTRONOMY 

arises, what is the distance of the nearest fixed star? What 
is the scale on which our visible firmament is constructed? 
And what proportion do its dimensions bear to those of our 
own immediate system? To these questions astronomy has 
at length been enabled to afford an answer. 

(800.) The diameter of the earth has served us for the 
base of a triangle, in the trigonometrical survey of our system 
(art. 274), by which to calculate the distance of the sun; 
but the extreme minuteness of the sun's parallax (art. 357) 
renders the calculation from this "ill-conditioned" triangle 
(art. 275) so delicate, that nothing but the fortunate com- 
bination of favorable circumstances, afforded by the transits 
of Yenus (art. 479), could render its results even tolerably 
worthy of reliance. Bat the earth's diameter is too small 
a base for direct triangulation to the verge even of our own 
system (art. 526), and we are, therefore, obliged to substi- 
tute the annual parallax for the diurnal, or, which comes 
to the same thing, to ground our calculation on the relative 
velocities of the earth and planets in their orbits (art. 486), 
when we would push our triaogulation to that extent. It 
might be naturally enough expected, that by this enlarge- 
ment of our base to the vast diameter of the earth's orbit, 
the next step in our survey (art. 275) would be made at 
a great advantage; — that our change of station, from side 
to side of it, would produce a considerable and easily 
measurable amount of annual parallax in the stars, and 
that by its means we should come to a knowledge of their 
distance. But, after exhausting every refinement of obser- 
vation, astronomers were, up to a very late period, unable 
to come to any positive and coincident conclusion upon this 
head; and the amount of such parallax, even for the nearest 
fixed star examined with the requisite attention, remained 



OUTLINES OF ASTRONOMY 717 

mixed up with, and concealed among, the errors incidental 
to all astronomical determinations. The nature of these 
errors has been explained in the earlier part of this work, 
and we need not remind the reader of the difficulties which 
must necessarily attend the attempt to disentangle an ele- 
ment not exceeding a few tenths of a second or at most 
a whole second from the host of uncertainties entailed on 
the results of observations by them: none of them indi- 
vidually perhaps of greater magnitude, but embarrassing 
by their number and fluctuating amount. Nevertheless, 
by successive refinements in instrument making, and by 
constantly progressive approximation to the exact knowl- 
edge of the uranographical corrections, that assurance has 
been obtained, even in the earlier years of the present 
century, viz. that no star visible in northern latitudes, to 
which attention had been directed, manifested an amount of 
parallax exceeding a single second of arc. It is worth while 
to pause for a moment to consider what conclusions would 
follow from the admission of a parallax to this amount. 

(801.) Eadius is to the sine of V as 206265 to 1. In this 
proportion then at least must the distance of the fixed stars 
from the sun exceed that of the sun from the earth. Again, 
the latter distance, as we have already seen (art. 357), ex- 
ceeds the earth's radius in the proportion of 23984 to 1. 
Taking therefore the earth's radius for unity, a parallax 
of 1" supposes a distance of 4947059760 or nearly five thou- 
sand millions of such units: and lastly, to descend to ordi- 
nary standards, since the earth's radius may be taken at 
4000 of our miles, we find 19788239040000 or about twenty 
billions of miles for our resulting distance. 

(802.) In such numbers the imagination is lost. The 
only mode we have of conceiving such intervals at all is 



718 OUTLINES OF ASTRONOMY 

by the time which it would require for light to traverse 
them. Light, as we know (art. 645), travels at the rate of 
a semidiameter of the earth's orbit in 8 m 13 8 *3. It would, 
therefore, occupy 206205 times this interval or 3 years and 
83 days to traverse the distance in question. Now as this 
is an inferior limit which it is already ascertained that 
even the brightest and therefore probably the nearest stars 
exceed, what are we to allow for the distance of those 
ianumerable stars of the smaller magnitudes which the 
telescope discloses to us! What for the dimensions of 
the galaxy in whose remoter regions, as we have seen, the 
united lustre of myriads of stars is perceptible only in 
powerful telescopes as a feeble nebulous gleam! 

(803.) The space-penetrating power of a telescope or the 
comparative distance to which a given star would require to 
be removed in order that it may appear of the same bright- 
ness in the telescope as before to the naked eye, may be cal- 
culated from the aperture of the telescope compared with that 
of the pupil of the eye, and from its reflecting or transmitting 
power, i.e. the proportion of the incident light it conveys to 
the observer's eye. Thus it has been computed that the 
space-penetrating power of such a reflector as that used in 
the star-gauges above referred to is expressed by the num- 
ber 75. A star then of the sixth magnitude removed to 75 
times its distance would still be perceptible as a star with 
that instrument, and admitting such a star to have 100th 
part of the light of a standard star of the first magnitude, 
it will follow that such a standard star, if removed to 750 
times its distance, would excite in the eye, when viewed 
through the gauging telescope, the same impression as a star 
of the sixth magnitude does to the naked eye. Among the 
infinite multitude of such stars in the remoter regions of 



OUTLINES OF ASTRONOMY 719 

the galaxy, it is but fair to conclude that innumerable indi- 
viduals equal in intrinsic brightness to those which imme- 
diately surround us must exist. The light of such stars then 
must have occupied upward of 2000 years in travelling over 
the distance which separates them from our own system. 
It follows then that when we observe the places and note 
the appearances of such stars, we are only reading their his- 
tory of two thousand years' anterior date thus wonderfully 
recorded. We cannot escape this conclusion but by adopt- 
ing as an alternative an intrinsic inferiority of light in all 
the smaller stars of the galaxy. We shall be better able to 
estimate the probability of this alternative when we shall 
have made acquaintance with other sidereal systems whose 
existence the telescope discloses to us, and whose analogy 
will satisfy us that the view of the subject here taken is in 
perfect harmony with the general tenor of astronomical facts. 
(804.) Hitherto we have spoken of a parallax of 1" as a 
mere limit below which that of any star yet examined as- 
suredly, or at least very probably falls, and it is not without 
a certain convenience to regard this amount of parallax as a 
sort of unit of reference, which, connected in the reader's 
recollection with a parallactic unit of distance from our sys- 
tem of 20 billions of miles, and with a %\ year's journey of 
light, may save him the trouble of such calculations, and 
ourselves the necessity of covering our pages with such 
enormous numbers, when speaking of stars whose parallax 
has actually been ascertained with some approach to cer- 
tainty, either by direct meridian observation or by more 
refined and delicate methods. These we shall proceed to 
explain, after first pointing out the theoretical peculiarities 
which enable us to separate and disentangle its effects from 
those of the uranographical corrections, and from other 



720 OUTLINES OF ASTRONOMY 

causes of error which being periodical in their nature add 
greatly to the difficulty of the subject. The effects of pre- 
cession and proper motion (see art. 852) which are uniformly 
progressive from year to year, and that of nutation which 
runs through its period in nineteen years, it is obvious 
enough, separate themselves at once by these characters 
from that of parallax; and, being known with very great 
precision, and being certainly independent, as regards their 
causes, of any individual peculiarity in the stars affected by 
them, whatever small uncertainty may remain respecting 
the numerical elements which enter into their computation 
(or in mathematical language their coefficients), can give rise 
to no embarrassment. With regard to aberration the case is 
materially different. This correction affects the place of a 
star by a fluctuation annual in its period, and therefore, so 
far, agreeing with parallax. It is also very similar in the 
law of its variation at different seasons of the year, parallax 
having for its apex (see arts. 343, 344) the apparent place of 
the sun in the ecliptic, and aberration a point in the same 
great circle 90° behind that place, so that in fact the for* 
mulse of calculation (the coefficients excepted) are the same 
for both, substituting only for the sun's longitude in the 
expression for the one, that longitude diminished by 90° 
for the other. Moreover, in the absence of absolute certainty 
respecting the nature of the propagation of light, astrono- 
mers have hitherto considered it necessary to assume at 
least as a possibility that the velocity of light may be to 
some slight amount dependent on individual peculiarities 
in the body emitting it. 12 



12 In the actual state of astronomy and photology this necessity can hardly 
be considered as still existing, and it is desirable, therefore, that the practice of 
astronomers of introducing an unknown correction for the constant of aberration 



OUTLINES OF ASTRONOMY 721 

(805.) If we suppose a line drawn from the star to the 
earth at all seasons of the year, it is evident that this line 
will sweep over the surface of an exceedingly acute, oblique 
cone, having for its axis the line joining the sun and star, 
and for its base the earth's annual orbit, which, for the 
present purpose, we may suppose circular. The star will 
therefore appear to describe each year about its mean place 
regarded as tixed, and in virtue of parallax alone, a minute 
ellipse, the section of this cone by the surface of the celes- 
tial sphere, perpendicular to the visual ray. But there is 
also another way in which the same fact may be represented. 
The apparent orbit of the star about its mean place as a cen- 
tre, will be precisely that which it would appear to describe, 
if seen from the sun, supposing it really revolved about that 
place in a circle exactly equal to the earth's annual orbit, in 
a plane parallel to the ecliptic. This is evident from the 
equality and parallelism of the lines and directions con- 
cerned. Now the effect of aberration (disregarding the 
slight variation of the earth's velocity in different parts 
of its orbit) is precisely similar in law, and differs only in 
amount, and in its bearing reference to a direction 90° dif- 
ferent in longitude. Suppose, in order to fix our ideas, the 
maximum of parallax to be 1* and that of aberration 20 *5", 
and let A B, a b, be two circles imagined to be described 
separately, as above, by the star about its mean place S, in 
virtue of these two causes respectively, y being a line paral- 
lel to that of the line of equinoxes. Then if in virtue of 
parallax alone, the star would be found at a in the smaller 
orbit, it would in virtue of aberration alone be found at A, 
in the larger, the angle a S A being a right angle. Draw- 

into their "equations of condition" for the determination of parallax, should be 
disused, since it actually tend3 to introduce error into the final result. 



722 



OUTLINES OF ASTRONOMY 



ing then A C equal and parallel to S a, and joining S C, it 
will in virtue of both simultaneously be found in C, i.e. m 
the circumference of a circle whose radius is S C, and at a 
point in that circle, in advance of A, the aberrational place, 
by the angle A S 0. Now since S A : A C : : 20-5 : 1, we 
find for the angle A S G 2° 47' 35", and for the length of the 
radius S C of the circle representing the compound motion 
20" -524. The difference (0"-024) between this and S C, the 
radius of the aberration circle, is quite imperceptible, and 
even supposing a quantity so minute to be capable of detec- 

C 




tion by a prolonged series of observations, it would remain 
a question whether it were produced by parallax or by a 
specific difference of aberration from the general average 
20" -5 in the star itself. It is therefore to the difference of 
2° 48' between the angular situation of the dispkeed'star iu 
this hypothetical orbit, i.e. in the arguments (as they are 
called) of the joint correction (j'SC) and that of aberration 
alone (/ S A), that we have to look for the resolution of the 
problem of parallax. The reader may easily figure to him- 
self the delicacy of an inquiry which turns wholly (even 
when stripped of all its other dime ul ties) on the precise de- 



OUTLINES OF ASTRONOMY 723 

i nation of a quantity of this nature, and of such, very 
moderate magnitude. 

(806.) But these other difficulties themselves are of no 
trifling order. All astronomical instruments are affected 
by differences of temperature. Not only do the materials 
of which they are composed expand and contract, but the 
masonry and solid piers on which they are erected, nay even 
the very soil on which these are founded, participate in the 
general change from summer warmth to winter cold. Hence 
arise slow oscillatory movements of exceedingly minute 
amount, which levels and plumb-lines afford but very in- 
adequate means of detecting, and which being also annual 
in their period (after rejecting whatever is merely casual and 
momentary) mix themselves intimately with the matter of 
our inquiry. Refraction too, besides its casual variations 
from night to night, which a long series of observations 
would eliminate, depends for its theoretical expression on 
the constitution of the strata of our atmosphere, and the 
law of the distribution of heat and moisture at different ele- 
vations, which cannot be unaffected by difference of season. 
No wonder then that mere meridional observations should, 
almost up to the present time, have proved insufficient, 
except in one very remarkable instance, to afford unques- 
tionable evidence, and satisfactory quantitative measure- 
ment of the parallax of any fixed star. 

(807.) The instance referred to is that of a Centauri, one 
of the brightest and for many other reasons, one of the most 
remarkable of the southern stars. From a series of obser- 
vations of this star, made at the Royal Observatory of the 
Cape of Good Hope in the years 1832 and 1833, by Professor 
Henderson, with the mural circle of that establishment, a 

parallax to the amount of an entire second was concluded 

Astronomy— Vol. XX— 12 



724 OUTLINES OF ASTRONOMY 

on his reduction of the observations in question after his 
return to England. Subsequent observations by Mr. Mac- 
lear, partly with the same, and partly with a new and far 
more efficiently constructed instrument of the same descrip- 
tion made in the years 1839 and 1840, have fully confirmed 
the reality of the parallax indicated by Professor Hender- 
son's observations, though with a slight diminution in its 
concluded amount, which comes out equal to 0' / '9128 or 
about 4?ths of a second; bright stars in its immediate neigh- 
borhood being unaffected by a similar periodical displacement, 
and thus affording satisfactory proof that the displacement in- 
dicated in the case of the star in question is not merely a result 
of annual variations of temperature. As it is impossible at 
present to answer for so minute a quantity as that by which 
this result differs from an exact second, we may consider 
the distance of this star as approximately expressed by the 
parallactic unit of distance referred to in art. 804. 

(808.) A short time previous to the publication 13 of this 
important result, the detection of a sensible and measurable 
amount of parallax in the star N° 61 Cygni of Flamsteed's 
catalogue of stars was announced by the celebrated astrono- 
mer of Kdnigsberg, the late M. Bessel. 14 This is a small 
and inconspicuous star, hardly exceeding the sixth magni- 
tude, but which had been pointed out for especial observa- 
tion by the remarkable circumstance of its being affected by 
a proper motion (see art. 852), i.e. a regular and continually 
progressive annual displacement among the surrounding 
stars to the extent of more than 5" per annum, a quantity 
so very much exceeding the average of similar minute an- 



13 Prof. Henderson's paper was read before the Astronomical Society of 
London, Jan. 3, 1839. It bears date Dec. 24, 1838. 

14 Astronomische Nackrichten, Nos. 365, 366, Dec. 13, 1838. 



OUTLINES OF ASTRONOMY 725 

nual displacements which many other stars exhibit, as to 
lead to a suspicion of its being actually nearer to our sys- 
tem. It is not a little remarkable that a similar presump- 
tion of proximity exists also in the case of a Centauri, whose 
unusually large proper motion of nearly 4" per annum is 
stated by Professor Henderson to have been the motive 
which induced him to subject his observations of that star 
to that severe discussion which led to the detection of its 
parallax. M. Bessel's observations of 61 Cygni were com 
menced in August, 1837, immediately on the establishment 
at the Konigsberg observatory of a magnificent heliometer, 
the workmanship of the celebrated optician Fraunhofer of 
Munich, an instrument especially fitted for the system of ob- 
servation adopted; which being totally different from that 
of direct meridional observation, more refined in its concep- 
tion, and susceptible of far greater accuracy in its practical 
application, we must now explain. 

(809.) Parallax, proper motion, and specific aberration 
(denoting by the latter phrase that part of the aberration of 
a star's light which may be supposed to arise from its indi- 
vidual peculiarities, and which we have every reason to be- 
lieve at all events an exceedingly minute fraction of the 
whole) are the only uranographical corrections which do 
not necessarily affect alike the apparent places of two stars 
situated in, or very nearly in, the same visual line. Sup- 
posing then two stars at an immense distance, the one be- 
hind the other, but otherwise so situated as to appear very 
nearly along the same visual line, they will constitute what 
is called a star optically double, to distinguish it from a star 
physically double, of which more hereafter. Aberration 
(that which is common to all stars), precession, nutation, 
nay, even refraction, and instrumental causes of apparent dis- 



726 OUTLINES OF ASTRONOMY 

placement, will affect them alike, or so very nearly alike (if 
the minute difference of their apparent places be taken into 
account) as to admit of the difference being neglected, or 
very accurately allowed for, by an easy calculation. If 
then, instead of attempting to determine by observation the 
place of the nearer of two very unequal stars (which will 
probably be the larger) by direct observation of its right 
ascension and polar distance, we content ourselves with re- 
ferring its place to that of its remoter and smaller com- 
panion by differential observation, i.e. by measuring only 
its difference of situation from the latter, we are at once 
relieved of the necessity of making these corrections, and 
from all uncertainty as to their influence on the result. 
And for the very same reason, errors of adjustment (art. 
136), of graduation, and a host of instrumental errors, 
which would for this delicate purpose fatally affect the 
absolute determination of either star's place, are harmless 
when only the difference of their places, each equally 
affected by such causes, is required to be known. 

(810.) Throwing aside therefore the consideration of all 
these errors and corrections, and disregarding for the pres- 
ent the minute effect of specific aberration and the uni- 
formly progressive effect of proper motion, let us trace the 
effect of the differences of the parallaxes of two stars thus 
juxtaposed, or their apparent relative distance and position 
at various seasons of the year. Now the parallax being 
inversely as the distance, the dimensions of the small el- 
lipses apparently described (art. 805) by each star on the 
concave surface of the heavens by parallactic displacement 
will differ — the nearer star describing the larger ellipse. 
But both stars lying very nearly in the same direction from 
the sun, these ellipses will be similar and similarly situated. 



OUTLINES OF ASTRONOMY 



727 



Suppose S and 5 to be the positions of the two stars as seen 
from the sun, and let A B C D, a b c c?, be their parallactic 
ellipses; then, since they will be at all times similarly situ- 
ated in these ellipses, when the one star is seen at A, the 
other will be seen at a. When the earth has made a quar- 
ter of a revolution in its orbit, their apparent places will be 
B &; when another quarter, C c; and when another, D d. 
If, then, we measure carefully, with micrometers adapted 




for the purpose, their apparent situation with respect to each 
other, at different times of the year, we should perceive a 
periodical change, both in the direction of the line joining 
them, and in the distance between their centres. For the 
lines A a and C c cannot be parallel, nor the lines B b and 
D d equal, unless the ellipses be of equal dimensions, i.e. 
unless the two stars have the same parallax, or are equidis- 
tant from the earth. 

(811.) Now, micrometers, properly mounted, enable us 
to measure very exactly both the distance between two ob- 
jects which can be seen together in the same field of a tele- 
scope, and the position of the line joining them with respect 
to the horizon, or the meridian, or any other determinate 
direction in the heavens. The double image micrometer, 
and especially the heliometer (arts. 200, 201), is peculiarly 



728 OUTLINES OF ASTRONOMY 

adapted for this purpose. The images of the two stars 
formed side by side, or in the same line prolonged, however 
momentarily displaced by temporary refraction or instru- 
mental tremor, move together, preserving their relative situa- 
tion, the judgment of which is no way disturbed by such 
irregular movements. The heliometer also, taking in a 
greater range than ordinary micrometers, enables us to 
compare one large star with more than one adjacent small 
one, and to select such of the latter among many near it, as 
shall be most favorably situated for the detection of any 
motion in the large one, not participated in by its neighbors. 
(812.) The star examined by Bessel has two such neigh- 
bors, both very minute, and therefore probably very distant, 
most favorably situated, the one (s) at a distance of T 42", 
the other (s') at 11' 46" from the large star, and so situated, 
that their directions from that star make nearly a right angle 
with each other. The effect of parallax therefore would 
necessarily cause the two distances S s and S s' to vary so 
as to attain their maximum and minimum values alternately 
at three-monthly intervals, and this is what was actually ob. 
served to take place, the one distance being always most 
rapidly on the increase or decrease when the other was sta- 
tionary (the uniform effect of proper motion being under- 
stood of course to be always duly accounted for). This al- 
ternation, though so small in amount as to indicate, as a 
final result, a parallax, or rather a difference of parallaxes 
between the large and small stars of hardly more than one- 
third of a second, was maintained with such regularity as to 
leave no room for reasonable doubt as to its cause, and hav- 
ing been confirmed by the further continuance of these ob- 
servations, and quite recently by the exact coincidence 
between the result thus obtained, and that deduced by M. 



OUTLINES OF ASTRONOMY 729 

Peters from observations of the same star at the observatory 
of Pulkova, 16 is considered on all hands as fully established. 
The parallax of this star finally resulting from Bessel's ob- 
servation is 0""348, so that its distance from our system is 
very nearly three parallactic units. (Art. 804.) 

(813.) The bright star a Lyras has also near it, at only 43" 
distance (and therefore within the reach of the parallel wire 
or ordinary double image micrometer), a very minute star, 
which has been subjected since 1835 to a severe and assidu- 
ous scrutiny by M. Struve, on the same principle of differ- 
ential observation. He has thus established the existence 
of a measurable amount of parallax in the large star, less 
indeed than that of 61 Cygni (being only about J of a sec- 
ond), but yet sufficient (such was the delicacy of his meas- 
urements) to justify this excellent observer in announcing 
the result as at least highly probable, on the strength of only 
five nights' observation, in 1835 and 1836. This probabil- 
ity, the continuation of the measures to the end of 1838 and 
the corroborative, though not in this case precisely coinci- 
dent, result of M. Peters's investigations have converted 
into a certainty. M. Struve has the merit of being the first 
to bring into practical application this method of observa- 
tion, which, though proposed for the purpose, and its great 
advantages pointed out by Sir William Herschel so early as 
1781, 18 remained long unproductive of any result, owing 
partly to the imperfection of micrometers for the measure- 
ment of distance, and partly to a reason which we shall pres- 
ently have occasion to refer to. 



15 "With the great vertical circle by Ertel. 

16 It has been referred even to Galileo. But the general explanation of 
Parallax in the Systema Cosmicum, Dial. iii. p. 271 (Leyden edit. 1699), to which 
the reference applies, does not touch any of the peculiar features of the case, 
or meet any of its difficulties. 



730 OUTLINES OF ASTRONOMY 

(814.) If the component individuals S, s {fig. art. 810) be 
(as is often the case) very close to each other, the parallactic 
variation of their angle of position, or the extreme angle in- 
cluded between the lines A a, C c, may be very consider- 
able, even for a small amount of difference of parallaxes 
between the large and small stars. For instance, in the case 
of two adjacent stars 15" asunder, and otherwise favorably 
situated for observation, an annual fluctuation to and fro in 
the apparent direction of their line of junction to the extent 
of half a degree (a quantity which could not escape notice in 
the means of numerous and careful measurements) would 
correspond to a difference of parallax of only J of a second, 
A difference of 1" between two stars apparently situated at 
5" distance might cause an oscillation in that line to the ex- 
tent of no less than 11°, and if nearer one proportionally 
still greater. This mode of observation has been applied to 
a considerable number of stars by Lord Wrottesley, and 
with such an amount of success, as to make its further 
application desirable. (Phil. Trans. 1851.") 

(815.) The following are some of the principal fixed stars 
to which parallax has been up to the present time more or 
less probably assigned: 



^ 



« Centauri 0*976 (Henderson, corrected by Peters.) 

61 Cjgni 0-348 (Bessel.) 

Lalande 21258. . . 0-260 (Kriiger.) 

Oeltzen . 17415-6 . . 0-247 (Kriiger.) 

« Lyrse 0*155 (W. Struve, corrected by 0. Slruve.) 

Sirius 0-150 (Henderson, corrected by Peters.) 

10 p Ophiuchi 0-16 (Kriiger.) 

* TJrsse Majoris .... 0*139 (Peters.) 

Arcturus 0-127 ditto 

Polaris 0-067 ditto 

Capella 0-046 ditto 

17 See Phil. Trans. 1826, p. 266 et seq., and 1827, for a list of stars we'd 
adapted for such observation, with the times of the year most favorable. — Th9 
list in Phil. Trans. 1826 is incorrect. 



OUTLINES OF ASTRONOMY 731 

Although the extreme minuteness of the last four of these 
results deprives them of much numerical reliance, it is at 
least certain that the parallaxes by no means follow the 
order of magnitudes, and this is further shown by the fact 
that a Cygni, one of M. Peters's stars, shows absolutely no 
indications of any measurable parallax whatever. 

(816.) From the distance of the stars we are naturally 
led to the consideration of their real magnitudes. But here 
a difficulty arises, which, so far as we can judge of what 
optical instruments are capable of effecting, must always 
remain insuperable. Telescopes afford us only negative in- 
formation as to the apparent angular diameter of any star. 
The round, well-defined, planetary disks which good tel- 
escopes show when turned upon any of the brighter stars 
are phenomena of diffraction, dependent, though at present 
somewhat enigmatically, on the mutual interference of the 
rays of light. They are consequently, so far as this inquiry 
is concerned, mere optical illusions, and have therefore been 
termed spurious disks. The proof of this is that telescopes 
of different apertures and magnifying powers, when applied 
for the purpose of measuring their angular diameters, give 
different results, the greater aperture (even with the same 
magnifying power) giving the smaller disk. That the true 
disk of even a large and bright star can have but a very 
minute angular measure, appears from the fact that in the 
occultation of such a star by the moon, its extinction is 
absolutely instantaneous, not the smallest trace of gradual 
diminution of light being perceptible. The apparent or 
spurious disk also remains perfectly round and of its full 
size up to the instant of disappearance, which could not 
be the case were it a real object. If our sun were removed 
to the distance expressed by our parallactic unit (art. 804), 



732 OUTLINES OF ASTRONOMY 

its apparent diameter of 32' 1"*5 would be reduced to only 
0"*0093, or less than the hundredth of a second, a quantity 
which we have not the smallest reason to hope any practical 
improvement in telescopes will ever show as an object hav- 
ing distinguishable form. 

(817.) There remains therefore only the indication which 
the quantity of light they send to us may afford. But here 
again another difficulty besets us. The light of the sun is 
so immensely superior in intensity to that of any star, that 
it is impracticable to obtain any direct comparison between 
them. But by using the moon as an intermediate term of 
comparison it may be done, not indeed with much precision, 
but sufficiently well to satisfy in some degree our curiosity 
on the subject. Now a Centauri has been directly compared 
with the moon by the method explained in art. 783. By 
a mean of eleven such comparisons made in various states 
of the moon, duly reduced and making the proper allowance 
on photometric principles for the moon's light lost by trans- 
mission through the lens and prism, it appears that the mean 
quantity of light sent to the earth by a full moon exceeds 
that sent by a Centauri in the proportion of 27408 to I. 
Now Wollaston, by a method apparently unobjectionable, 
found 18 the proportion of the sun's light to that of the full 
moon to be that of 801072 to 1. Combining these results, 
we find the light sent us by the sun to be to that sent by 
a Centauri as 21,955,000,000, or about twenty-two thousand 
millions to 1. Hence from the parallax assigned above to 
that star, it is easy to conclude that its intrinsic splendor, 
as compared with that of our sun at equal distances, is 
2*3247, that of the sun being unity." 

18 Wollaston, Phil. Trans. 1829, p. 27. 

19 Results of Astronomical Observations at the Cape of Good Hope, etc. , art. 



OUTLINES OF ASTRONOMY 733 

(818.) The light of Sirius is four times that of a Centauri, 
and its parallax only 0"-15. (Art. 230.) This in effect as- 
cribes to it an intrinsic splendor equal to 169*35 times that 
of a Centauri, and therefore 393*7 times that of our sun. 80 



CHAPTER XVI 

Variable and Periodical Stars — List of those Already Known — Irregulari- 
ties in their Periods and Lustre when Brightest — Irregular and Tem- 
porary Stars — Ancient Chinese Records of Several — Missing Stars — 
Double Stars — Their Classification — Specimens of each Class — Binary 
Systems — Revolution Round each other — Describe Elliptic Orbits under 
the Newtonian Law of Gravity — Elements of Orbits of Several — Actual 
Dimensions of their Orbits — Colored Double Stars — Phenomenon of 
Complementary Colors — Sanguine Stars — Proper Motion of the Stars 
—Partly Accounted for by a Real Motion of the Sun — Situation of 
the Solar Apex — Agreement of Southern and Northern Stars in Giving 
the Same Result — Principles on which the Investigation of the Solar 
Motion Depends — Absolute Telocity of the Sun's Motion — Supposed 
Revolution of the "Whole Sidereal System Round a Common Centre 
— Systematic Parallax and Aberration — Effect of the Motion of Light 
in Altering the Apparent Period of a Binary Star 

(819.) Now, for what purpose are we to suppose such 
magnificent bodies scattered through the abyss of space? 
Surely not to illuminate our nights, which an additional 
moon of the thousandth part of the size of our own would 

278, p. 363. If only the results obtained near the quadratures of the moon 
(which is the situation most favorable to exactness) be used, the resulting value 
of the intrinsic light of the star (the sun being unity) i3 4'1586. On the other 
hand, if only those procured near the full moon (the worst time for observation) 
be employed, the result is 1 # 4017. Discordances of this kind will startle no one 
conversant with photometry. That « Centauri really emits more light than our 
sun must, we conceive, be regarded as an established fact. To those who may 
refer to the work cited it is necessary to mention that the quantity there desig- 
nated by M, expresses, on the scale there adopted, 500 times the actual illumi- 
nating power of the moon at the time of observation, that of the mean full moon 
being unity. 

20 See the work above cited, p. 367. — Wollaston makes the light of Sirius 
one 20,000-millionth of the sun's. Steinheil by a very uncertain method found 
— (3286500) 3 X Arcturus. 



73± OUTLINES OF ASTRONOMY 

do much better, nor to sparkle as a pageant void of mean- 
ing and reality, and bewilder us among vain conjectures. 
Useful, it is true, they are to man as points of exact and 
permanent reference; but he must have studied astronomy 
to little purpose, who can suppose man to be the only object 
of his Creator's care, or who does not see in the vast and 
wonderful apparatus around us provision for other races 
of animated beings. The planets, as we have seen, derive 
their light from the sun; but that cannot be the case with 
the stars. These doubtless, then, are themselves suns, and 
may, perhaps, each in its sphere, be the presiding centre 
round which other planets, or bodies of which we can form 
no conception from any analogy offered by our own system, 
may be circulating. 

(820.) Analogies, however, more than conjectural, ar8 
not wanting to indicate a correspondence between the dy- 
namical laws which prevail in the remote regions of the 
stars and those which govern the motions of our own 
system. Wherever we can trace the law of periodicity — the 
regular recurrence of the same phenomena in the same 
times — we are strongly impressed with the idea of rotatory 
or orbitual motion. Among the stars are several which, 
though no way distinguishable from others by any apparent 
change of place, nor by any difference of appearance in 
telescopes, yet undergo a more or less regular periodical 
increase and diminution of lustre, involving in one or two 
cases a complete extinction and revival. These are called 
periodical stars. The longest known and one of the most 
remarkable is the star Omicron, in the constellation Cetus 
(sometimes called Mira Ceti), which was first noticed as 
variable by Fabricius in 1596. It appears about twelve 
times in eleven years, or more exactly in a period of 



OUTLINES OF ASTRONOMY 735 

33 1 J 8 h 4 m 16 8 ; remains at its greatest brightness about a 
fortnight, being then on some occasions equal to a large 
star of the second magnitude; decreases during about three 
months, till it becomes completely invisible to the naked 
eye, in which state it remains about five months: and con- 
tinues increasing daring the remainder of its period. Such 
is the general course of its phases. It does not always 
however return to the same degree of brightness, nor in- 
crease and diminish by the same gradations, neither are 
the successive intervals of its maxima equal. From the 
recent observations and inquiries into its history by M. 
Argelander, the mean period above assigned would appear 
to be subject to a cyclical fluctuation embracing eighty- 
eight such periods, and having the effect of gradually 
lengthening and shortening alternately those intervals to 
the extent of twenty-five days one way and the other. 1 
The irregularities in the degree of brightness attained at 
the maximum are probably also periodical. Hevelius re- 
lates 2 that during the four years between October, 1672, 
and December, 1676, it did not appear at all. It was un- 
usually bright on October 5, 1839 (the epoch of its maxi- 
mum for that year according to M. Argelander's observa- 
tions), when it exceeded a Ceti and equalled p Aurigse in 
lustre. When near its minimum its color changes from 
white to a full red. 

(821.) Another very remarkable periodical star is that 
called Algol, or (3 Persei. It is usually visible as a star 
of the second magnitude, and such it continues for the 
space of 2 d 13^, when it suddenly begins to diminish in 
splendor, and in about SI hours is reduced to the fourth 
magnitude, at which it continues about 15 m . It then begins 

1 Astronom. Nachr. No. 624. . 8 Lalande's Astronomy, Art. 794. 



736 OUTLINES OF ASTRONOMY 

again to increase, and in S\ hours more is restored to its 
usual brightness, going through all its changes in 2 d 20 h 48 m 
54 8 -7. This remarkable law of variation certainly appears 
strongly to suggest the revolution round it of some opaque 
body, which when interposed between us and Algol, cuts 
off a large portion of its light; and this is accordingly the 
view taken of the matter by Groodricke, to whom we owe 
the discovery of this remarkable fact, 3 in the year 1782; 
since which time the same phenomena have continued to 
be observed, but with this remarkable additional point of 
interest; viz. that the more recent observations as compared 
with the earlier ones indicate a diminution in the periodic 
time. The latest observations of Argelander, Heis and 
Schmidt even go to prove that this diminution is not uni- 
formly progressive, but is actually proceeding with acceler- 
ated rapidity, which however will probably not continue, 
but, like other cyclical combinations in astronomy, will by 
degrees relax, and then be changed into an increase, accord- 
ing to laws of periodicity which, as well as their causes, 
remain to be discovered. The first minimum of this star in 
the year 1844 occurred on January 3, at 4 h 14 m Greenwich 
mean time. 4 

(822.) The star 8 in the constellation Cepheus is also 
subject to periodical variations, which, from the epoch of- 
its first observation by Groodricke in 1784 to the present 

3 The same discovery appears to have been made nearly about the same 
time by Palitzch, a farmer of Prolitz, near Dresden — a peasant by station, an 
astronomer by nature — who, from his familiar acquaintance with the aspect of 
the heavens, had been led to notice among so many thousand stars this one 
as distinguished from the rest by its variation, and had ascertained its period. 
The same Palitzch was also the first to rediscover the predicted comet of Halley 
in IT 59, which he saw nearly a month before any of the astronomers, who, 
armed with their telescopes, were anxiously watching its return. These anec- 
dotes carry us back to the era of the Chaldean shepherds. Montanari in 1669, 
and Maraldi in 1694, had already noticed a fluctuation of brightness in Algol. 

4 Ast. Nach. No. 472. 



OUTLINES OF ASTRONOMY 737 

time, have been continued with perfect regularity. Its 
period from minimum to minimum is 5 d 8 h 47 m 39 8, 5, the 
first or epochal minimum for 1849 falling, on January 2, 
3 h 13 m 37 s M. T. at Greenwich. The extent of its variation 
is from the fifth to between the third and fourth magni- 
tudes. Its increase is more rapid than its diminution, the 
interval between the minimum and maximum of its light 
being only l d 14 h , while that from the maximum to the 
minimum is 3 d 19 h . 

(823.) The periodical star p Lyrse, discovered by Good- 
ricke also in 1784, has a period which has been usually 
stated at from 6 d 9 h to 6 d ll h , and there is no doubt that in 
about this interval of time its light undergoes a remarkable 
diminution and recovery. The more accurate observations 
of M. Argelander however have led him to conclude 6 the 
true period to be 12 d 21 h 53 m 10 s , and that in this period 
a double maximum and minimum takes place, the two 
maxima being nearly equal and both about the 3*4 magni- 
tude, but the minima considerably unequal, viz. 4*3 and 
4-5m. In addition to this curious subdivision of the whole 
interval of change into two semiperiods we are presented 
in the case of this star with another instance of slow altera- 
tion of period, which has all the appearance of being itself 
periodical. From the epoch of its discovery in 1784 to the 
year 1840 the period was continually lengthening, but more 
and more slowly, till at the last-mentioned epoch it ceased 
to increase, and has since been slowly on the decrease. As 
an epoch for the least or absolute minimum of this star, 
M. Argelander's calculations enable us to assign 1846 Jan- 
uary 3 d h 9 m 53 s G. M. T. 

5 Astron. N"achr. No. 624. See also the valuable papers by this excellent 
astronomer in A. N. Nos. 417. 455, etc. 



738 



OUTLINES OF ASTRONOMY 



(824.) The following list comprises most of the variable 
stars at present known: 













Mas-n. 






Star. 


R. A. 

1850. 


N. P. D. 
1850. 






Period, 
Days. 


Discovered by 

















Max. 


Min. 






Il 


m 














T? Pise. . 





24 


76° 


23' 


9-5 


11 


242-L. 


Luther, 1855. 


* Cass. . 





32 


34 


17 


2 


2-5 


79-1 


Birt, 1831. 


S? Pise. . 


1 


10 


81 


51 


9 


13 




Hind, 1851. 


R ? Pise. . 


1 


23 


87 


54 


7-5 


9-5 


343 


Hind, 1850. 


o Celi 


2 


12 


93 


40 


2 


12 


331-336 


Fabricius, 1596. 


J8 Pers. . 


2 


58 


49 


38 


2-3 


4-5 


2-8673 


Goodricke, 1782. 


A Taur. . 


3 


52 


77 


56 


4 


5-4 


4± 


Baxendell, 1848. 


R? Taur. . 


4 


20 


80 


10 


8 


13-5 




Hind, 1849. 


S? Taur. . 


4 


22 


80 


22 


8-0 


12-5 


' 257 ' 




R? Orio. . 


4 


51 


82 


6 


9 


12-5 


237? 


Hind, 1848. 


6 Ami. . 


4 


51 


46 


24 


3 


4 


250-L. 


Heis, 1846. 




4 


53 


105 


2 


7 






Schmidt, 1855. 


a Orio. . 


5 


47 


82 


38 


1 


1-5 




J. Herschel, 1836. 


' G-emi. . 


6 


55 


69 


13 


3-7 


4-5 


10-15 


Schmidt, 1847. 


R? Gemi. . 


6 


58 


67 


4 


7 


11 


370 


Hind, 1848. 


R ? Cau. m. 


7 





79 


44 


8 






Argelander, 1854. 


S ? Can. m. 


7 


25 


81 


22 


8-1 






Hind, 1856. 


S? Gemi. . 


7 


34 


66 


12 


9 


13-5 


295 


Hind, 1848. 


T? Gemi. . 


7 


40 


65 


54 


9 


13-5 


287 


Hind, 1848. 


U? Gemi. . 


7 


46 


67 


37 


9 


13-5 


100 V 


Bind, 1855. 


R? Cane. . 


8 


8 


77 


51 


6 


10 


380 


Schwerd, 1829. 


S? Cane. . 


8 


35 


70 


25 


8 


10-5 


9-484 


Hind, 1848. 


S? Hyd. . 


8 


46 


S6 


22 


8-5 


13-5 


260 


Hind, 1848. 




8 


48 


69 


35 


85 


12 




Hind, 1850. 


T? Hyd. . 


8 


49 


98 


39 


8-5 


10-5 


240+ 


Hind, 1851. 


a Hyd. . 


9 


20 


98 





2-5 


3 


55 


J. Herschel, 1837. 


3 Leon. . 


9 


21 


81 


10 


6 





78 


Smyth,—? 


$ Leon. . 


9 


36 


75 


18 


6 





Long. 


Montanari, 1667. 


R? Leon. 


9 


39 


77 


53 


5 


10 


313? 


Koch, 1782. 


R ? Urs. U. 


10 


34 


20 


26 


7-5 


13 


301-35 


Pogson, 1853. 


v Argus . 


10 


39 


148 


54 


1 


4 


46 years ? 


Burchell, 1823. 


a Urs. M. 


10 


54 


27 


26 


1-5 


2 


Long. 


Lalande, 1786. 


R? Comas. 


11 


57 


70 


23 


8-0 








« Urs. M. 


12 


8 


32 


8 


2 


2-5 


Long. 




21 Yirg. . 


12 


26 


98 


28 










R? Yirg. . 


12 


31 


82 


11 


6-5 


11 


145-724 


Harding, 1809. 


S ? Urs. M. 


12 


37 


28 


5 


7 


12 


221-750 


Pogson, 1853. 


U ? Yirg. . 


12 


43 


83 


38 


7-8 








v Hydr. . 


13 


22 


112 


20 


4 


10* 


495* 


Maraldi, 1704. 


S? Yirg. . 


13 


25 


96 


25 


5-5 


11 


377? 


Hind, 1852. 


v Urs. M. 


13 


42 


39 


56 


1-5 


2 


Long. 


Lalande, 1786. 


Libr. . 


14 


45 


101 


45 


8 


9-5 




Schumacher, — ? 


P Urs. m. 


14 


51 


15 


14 


2 


2-5 


Long. 


Struve, 1838. 


5 Libras. 


14 


53 


97 


55 











OUTLINES OF ASTRONOMY 



739 













Masrn. 






Star. 


R. A. 

1850. 


N. P. D. 
1850. 






Period, 
Days. 


Discovered by 
















Max. 


Min. 








h 


m 














S? Serp. . 


15 


15 


75° 


9' 


8 


10 


367 


Harding, 1828. 


R? Cor. B. 


15 


42 


61 


23 


6 




323 


Pigott, 1795. 


R? Serp. . 


15 


44 


74 


24 


6-5 


10* 


359 


Harding, 1826. 


R ? Scorp. . 


16 


9 


112 


20 


9-0 






Chacornac, — ? 


S ? Scorp. . 


16 


9 


112 


20 


9 


12' 




Chacornac,1855. 


SO Mess. . 


16 


9 


112 


26 








Pogson, 1860. 


S? Ophi. . 


16 


26 


106 


52 


9-3 


13-5 


220 ?' 


Pogson, 1854. 


* Nova . 


16 


51 


102 


39 


4-5 


13-5 




Hind, 1848. 


R? Ophi. . 


16 


59 


105 


53 


8 


13 


396 ?' 


Pogson, 1853. 


a Here. . 


17 


8 


75 


26 


3-1 


3-7 


66-33? 


W.Herschel,1795. 


" Cor. A. 


18 


23 


128 


50 


3 


6 


Long. 


Halley, 1676. 


R? Scut. . 


18 


39 


95 


51 


5 


9 


61 


Pigott, 1795. 


Lyrse . 


18 


45 


56 


49 


3-5 


4-5 


12-914 


Goodricke, 1784. 


13 Lyras . 


18 


51 


46 


15 


4-3 


4-6 


48 


Baxendell, 1856. 


R ? Aqui. . 


18 


59 


82 





6-5 








R ? Cygn. . 


19 


33 


40 


8 


8 


14' 


415-50 


Pogson, 1812. 


X Cygn. . 


19 


45 


57 


28 


5 


11 


406-06 


Kirch, 1687. 


1 Aqui- • 


19 


45 


89 


22 


3-3 


4-7 


7-1763 


Pigott, 1784. 


Cygn. . 


19 


51 


55 


19 


4-5 


5-5 


Long. 


J.Herschel, 1842. 


R? Capr. . 


20 


3 


104 


42 


9-5 


13-5 




Hind, 1848. 


34 Cygn. . 


20 


12 


52 


26 


3' 


6 


18 years ? 


Janson, 1600. 


Pelph. 


20 


33 


77 


49 


8 


8— 


274? 


Struve, 1823. 


Urs.m. 


20 


38 


1 


20 


5 


11 




Pogson, 1853. 


Aquar . 


20 


42 


95 


42 








Goldschmidt. 


T? Capr. . 


21 


15 


105 


47 


90 




' 274* 




BAG 7582 . 


21 


39 


31 


54 


3 


' 6 ' 


Long. 


W.Herschel,l782. 


S? Pega. . 


22 


15 


82 


44 


8-5 


13-5 




Hind, 1848. 


s Ceph. . 


22 


24 


32 


21 


3-7 


4-7 


5-3664 


Goodricke, 1784. 


/3 Pega. . 


22 


57 


62 


44 


2 


2-5 


41 


Schmidt, 1848. 


R? Pega. . 


22 


59 


80 


16 


8-5 


13-5 


350 


Hind, 1848. 


R? Aqua. 


23 


37 


106 


6 


6-5 


10 


388-50 


Harding, 1810. 


R? Cass. . 


23 


51 


39 


26 


6 


14 


434? 


Pogson, 1853. 



(826.) Irregularities similar to those which have been 
noticed in the case of o Ceti, in respect of the maxima and 
minima of brightness attained in successive periods, have 
been also observed in several others of the stars in the fore- 
going list. % Cjgni, for example, is stated by Cassini to 
have been scarcely visible throughout the years 1699, 1700, 
1701, at those times when it was expected to be most con- 
spicuous. No. 59 Scuti is sometimes visible to the naked 



740 OUTLINES OF ASTRONOMY 

eye at its minimum, and sometimes not so, and its maximum 
is also very irregular. Pigott's variable star in Corona is 
stated by M. Argelander to vary for the most part so little 
that the unaided eye can hardly decide on its maxima and 
minima, while yet after the lapse of whole years of these 
slight fluctuations, they suddenly become so great that the 
star completely vanishes. The variations of a Orionis, which 
were most striking and unequivocal in the years 1836-1840, 
within the years since elapsed became much less conspicu- 
ous. In January, 1849, they had recommenced; and on 
December 5, 1852, Mr. Fletcher observed a Orionis brighter 
than Capella, and actually the largest star in the Northern 
hemisphere. The star called U Greminorum, in the list above 
given, is stated by Mr. Pogson to be subject to alternations 
or twinklings of light from the ninth to the thirteenth mag- 
nitude, in intervals from nine to fifteen seconds, neighboring 
stars of equal brightness remaining steady ! 

(827.) These irregularities prepare us for other phenom- 
ena of stellar variation, which have hitherto been reduced 
to no law of periodicity, and must be looked upon, in rela- 
tion to our ignorance and inexperience, as altogether casual; 
or, if periodic, of periods too long to have occurred more 
than once within the limits of recorded observation. The 
phenomena we allude to are those of Temporary Stars, 
which have appeared, from time to time, in different parts 
of the heavens, blazing forth with extraordinary lustre; 
and after remaining awhile apparently immovable, have 
died away, and left no trace. Such is the star which, 
suddenly appearing some time about the year 125 B.C., and 
which was visible in the daytime, is said to have attracted 
the attention of Hipparchus, and led him to draw up a cata- 
logue of stars, the earliest on record. Such, too, was the 



OUTLINES OF ASTRONOMY 741 

star which appeared, A.D. 389, near a Aquilae, remaining 
for three weeks as bright as Venus, and disappearing en- 
tirely. In the years 945, 1264, and 1572, brilliant stars ap- 
peared in the region of the heavens between Cepheus and 
Cassiopeia; and, from the imperfect account we have of the 
places of the two earlier, as compared with that of the last, 
which was well determined, as well as from the tolerably 
near coincidence of the intervals of their appearance, we 
may suspect them, with Groodricke, to be one and the same 
star, with a period of 312 or perhaps of 156 years. The 
appearance of the star of 1572 was so sudden, that Tycho 
Brahe, a celebrated Danish astronomer, returning one even- 
ing (the 11th of November) from his laboratory to his 
dwelling-house, was surprised to find a group of country 
people gazing at a star, which he was sure did not exist 
half an hour before. This was the star in question. It was 
then as bright as Sirius, and continued to increase till it 
surpassed Jupiter when brightest, and was visible at mid- 
day. It began to diminish in December of the same year, 
and in March, 1574, had entirely disappeared. So, also, 
on the 10th of October, 1604, a star of this kind, and not 
less brilliant, burst forth in the constellation of Serpentarius, 
which continued visible till October, 1605. 

(828.) Similar phenomena, though of a less splendid 
character, have taken place more recently, as in the case 
of the star of the third magnitude discovered in 1670, by 
Anthelm, in the head of the Swan; which, after becoming 
completely invisible, reappeared, and, after undergoing one 
or two singular fluctuations of light, during two years, at 
last died away entirely, and has not since been seen. 

(829.) On the night of the 28th of April, 1848, Mr. Hind 
observed a star of the fifth magnitude or 5*4 (very conspicu- 



742 OUTLINES OF ASTRONOMY 

ous to the naked eye) in a part of the constellation Ophiu- 
chus (K.A. 16 h 51 m l 8 -5, N. P. D. 102° 39' 14"), where, from 
perfect familiarity with that region, he was certain that up 
to the 5th of that month no star so bright as 9'10m. "previ- 
ously existed. Neither has any record been discovered of a 
star being there observed at any previous time. From the 
time of its discovery it continued to diminish, without any 
alteration of place, and before the advance of the season 
rendered further observation impracticable, was nearly ex- 
tinct. Its color was ruddy, and was thought by many ob- 
servers to undergo remarkable changes, an effect probably 
of its low situation. 

(830.) The alterations of brightness in the southern star 
Y) Argus, which have been recorded, are very singular and 
surprising. In the time of Halley (1677) it appeared as a 
star of the fourth magnitude. Lacaille, in 1751, observed 
it of the second. In the interval from 1811 to 1815, it was 
again of the fourth; and again from 1822 to 1826 of the sec- 
ond. On the 1st of February, 1827, it was noticed by Mr. 
Burchell to have increased to the first magnitude, and to 
equal a Crucis. Thence again it receded to the second; and 
so continued until the end of 1837. All at once in the be- 
ginning of 1838 it suddenly increased in lustre so as to sur- 
pass all the stars of the first magnitude except Sirius, Cano- 
pus, and a Centauri, which last star it nearly equalled. 
Thence it again diminished, but this time not below the 
1st magnitude until April, 1843, when it had again in- 
creased so as to surpass Canopus, and nearly equal Sirius 
in splendor. In May, 1863 [as well as in the years 1866-68], 
according to Mr. Abbott [and Mr. John Tebbutt, Jr.], it 
was only of the 6th magnitude. 6 Professor Loomis consid- 

6 Notices of R. Ast. Soc. xxiv. p. 6; xxv. p. 192; xxviii. p. 200, and p. 266. 



OUTLINES OF ASTR0N03II 743 

ers it as periodical, the interval of the minima being about 
seventy years. 7 "A strange field of speculation," it has 
been remarked, "is opened by this phenomenon. The tem- 
porary stars heretofore recorded have all become totally 
extinct. Variable stars, so far as they have been carefully 
attended to, have exhibited periodical alternations, in some 
degree at least regular, of splendor and comparative obscur- 
ity. But here we have a star fitfully variable to an as- 
tonishing extent, and whose fluctuations are spread over 
centuries, apparently in no settled period, and with no regu- 
larity of progression. What origin can we ascribe to these 
sudden flashes and relapses? What conclusions are we to 
draw as to the habitability of a system depending for its 
supply of light and heat on so uncertain a source?" Specu- 
lations of this kind can hardly be termed visionary, when 
we consider that, from what has before been said, we are 
compelled to admit a community of nature between the 
fixed stars and our own sun; and reflect that geology testi- 
fies to the fact of extensive changes having taken place at 
epochs of the most remote antiquity in the climate and tem- 
perature of our globe difficult to reconcile with the operation 
of secondary causes, such as a different distribution of sea 
and land, but which would find an easy and natural ex- 
planation in a slow variation of the supply of light and heat 
afforded by the sun itself. 

(831.) The Chinese annals of Ma-touan-lin, 8 in which 
stand officially recorded, though rudely, remarkable astro- 
nomical phenomena, supply a long list of "strange stars," 
among which, though the greater part are evidently comets, 
some may be recognized as belonging in all probability to 



7 Notices of Royal Astr. Soc. xxix. p. 298. 

8 Translated by M. Edward Biot, Connoissance des Temps, 1846. 



T44 OUTLINES OF ASTRONOMY 

the class of Temporary Stars as above characterized. Such 
is that which is recorded to have appeared in A.D. 173, be- 
tween a and Centauri, which (no doubt, scintillating from 
its low situation) exhibited "the five colors," and remained 
visible from December in that year till July in the next. 
And another which these annals assign to A.D. 1011, and 
which would seem to be identical with a star elsewhere 
referred to A.D. 1012, "which was of extraordinary bril- 
liancy, and remained visible in the southern part of the 
heavens during three months," 9 a situation agreeing with 
the Chinese record, which places it low in Sagittarius. 
Among several less unequivocal is one referred to B.C. 
134, in Scorpio, which may possibly have been Hippar- 
chus's star. [Lastly, on May 12, 1866, a star of the second 
magnitude was unexpectedly noticed by Mr. Birmingham 
(at Tuam) near e Coronse. It diminished rapidly, having 
been seen by Mr. Huggins on May 15, 16, 17, 18, 19, 20 
respectively as 3*6, 4*2, 4*9, 5*3, 5-7, and 6*2m. After 
dwindling to 10m. it again recovered so far as to have 
been seen on October 5 by M. Schmidt as 7m. Its place 
for 1866 was E. A. 16 h 54 m ; N. P. D. 63° 42'. Its spectrum 
was twofold, exhibiting both positive and negative lines, in- 
dicating at once the presence of flame and absorptive vapors.] 
(832.) On a careful re-examination of the heavens, and 
a comparison of catalogues, many stars are now found to be 
missing; and although there is no doubt that these losses 
have arisen in the great majority of instances from mistaken 
entries, and in some from planets having been mistaken for 
stars, yet in some it is equally certain that there is no mis- 



9 Hind, Notices of the Astronomical Society, viii. 156, citing Hepidannus. 
He places the Chinese star of 173 A.D. between « and £ Canis Minoris, but 
M. Biot distinctly says, «, Z 3 -pied oriental du Centaur e. 



OUTLINES OF ASTRONOMY 745 

take in the observation or entry, and that the star has really 
been observed, and as really has disappeared from the 
heavens. The whole subject of variable stars is a branch 
of practical astronomy which has been too little followed 
up, and it is precisely that in which amateurs of the science, 
and especially voyagers at sea, provided with only good 
eyes, or moderate instruments, might employ their time to 
excellent advantage. Catalogues of the comparative bright- 
ness of the stars in each constellation have been constructed 
by Sir Wm. Herschel, with the express object of facilitat- 
ing these researches, and the reader will find them, and a 
full account of his method of comparison, in the Phil. Trans. 
1796, and subsequent years. 

(833.) We come now to a class of phenomena of quite a 
different character, and which give us a real and positive 
insight into the nature of at least some among the stars, and 
enable us unhesitatingly to declare them subject to the same 
dynamical laws, and obedient to the same power of gravita- 
tion, which governs our own system. Many of the stars, 
when examined with telescopes, are found to be double, i.e. 
to consist of two (in some cases three or more) individuals 
placed near together. This might be attributed to acciden- 
tal proximity, did it occur only in a few instances; but the 
frequency of this companionship, the extreme closeness, 
and, in many cases, the near equality of the stars so con- 
joined, would alone lead to a strong suspicion of a more 
near and intimate relation than mere casual juxtaposition. 
The bright star Castor, for example, when much magnified, 
is found to consist of two stars of nearly the third magni- 
tude, within 5" of each other. Stars of this magnitude, 
however, are not so common in the heavens as to render it 
otherwise than excessively improbable that, if scattered at 



746 OUTLINES OF ASTRONOMY 

random, they would fall so near. But this improbability 
becomes immensely increased by a consideration of the fact, 
that this is only one out of a great many similar instances. 
Michell, in 1767, applying the rules for the calculation of 
probabilities to the case of the six brightest stars in the 
group called the Pleiades, found the odds to be 500000 to 1 
against their proximity being the mere result of a random 
scattering of 1500 stars (which he supposed to be the total 
number of stars of that magnitude in the celestial sphere 10 ) 
over the heavens. Speculating further on this, as an indi- 
cation of physical connection rather than fortuitous assem- 
blage, he was led to surmise the possibility (since converted 
into a certainty, but at that time, antecedent to any observa- 
tion) of the existence of compound stars revolving about one 
another, or rather about their common centre of gravity. 
M. Struve, pursuing the same train of thought as applied 
specially to the cases of double and triple combinations of 
stars, and grounding his computations on a more perfect 
enumeration of the stars visible down to the 7th magnitude, 
in the part of the heavens visible at Dorpat, calculates that 
the odds are 125 to 1 against any two stars, from the 1st to 
the 7th magnitude inclusive, out of the whole possible num- 
ber of binary combinations then visible, falling (if fortui- 
tously scattered) within 4" of each other. Now the number 
of instances of such binary combinations actually observed 
at the date of this calculation was already 91, and many 
more have since been added to the list. Again, he calcu- 
lates that the odds against any such stars fortuitously scat- 
tered, falling within 32" of a third, so as to constitute a 



10 This number is considerably too small, and in consequence, Micbell's odds 
in this case materially overrated. But enough will remain, if this be rectified, 
fully to bear out his argument. Phil. Trans, vol. 57. 



OUTLINES OF ASTRONOMY 747 

triple star, is not less than 4340 to 1. Now, four such com- 
binations occur in the heavens; viz. Orionis, a Orionis, 11 
Monocerotis, and C Cancri. The conclusion of a physical 
connection of some kind or other is therefore unavoidable. 

(834.) Presumptive evidence of another kind is furnished 
by the following consideration. Both a Centauri and 61 
Oygoi are "Double Stars." Both consist of two individu- 
als, nearly equal, and separated from each other by an in- 
terval of about a quarter of a minute. In the case of 61 
Cygni, the stars exceeding the 7th magnitude, there is 
already a prima facie probability of 9 to 1 against their 
apparent proximity. The two stars of a Centauri are both 
at least of the 2d magnitude, of which altogether not more 
than about 50 or 60 exist in the whole heavens. But, waiv- 
ing this consideration, both these stars, as we have already 
seen, have a proper motion so considerable that, supposing 
the constituent individuals unconnected, one would speedily 
leave the other behind. Yet at the earliest dates at which 
they were respectively observed these stars were not per- 
ceived to be double, and it is only to the employment of 
telescopes magnifying at least 8 or 10 times, that we owe the 
knowledge we now possess of their being so. With such a 
telescope Lacaille, in 1751, was barely able to perceive the 
separation of the two constituents of a Centauri, whereas, 
had one of them only been affected with the observed proper 
motion, they should then have been 6' asunder. In these 
cases then some physical connection may be regarded as 
proved by this fact alone. 

(835.) Sir William Herschel has enumerated upward of 
500 double stars, of which the individuals are less than 32" 
asunder. M. Struve, prosecuting the inquiry with instru- 
ments more conveniently mounted for the purpose, and 

Astronomy — Vol. XX — 13 



748 OUTLINES OF ASTRONOMY 

wrought to an astonishing pitch of optical perfection, has 
added more than five times that number. And other ob- 
servers have extended still further the catalogue of "Double 
Stars," without exhausting the fertility of the heavens. 
Among these are a great many in which the distance be- 
tween the component individuals does not exceed a single 
second. They are divided into classes by M. Struve (the 
first living authority in this department of astronomy), ac- 
cording to the proximity of their component individuals. 
The first class comprises those only in which the distance 
does not exceed 1" ; the 2d those in which it exceeds 1" but 
falls short of 2"; the 3d class extends from 2" to 4" distance; 
the 4th from 4" to 8"; the 5th from 8" to 12" ; the 6th from 
12" to 16"; the 7th from 16" to 24", and the 8th from 24" to 
32". Each class he again subdivides into two sub-classes of 
which the one under the appellation of conspicuous double 
stars {Duplices lucidce) comprehends those in which both in- 
dividuals exceed the 8J magnitude, that is to say, are sepa- 
rately bright enough to be easily seen in any moderately 
good telescope. All others, in which one or both the con- 
stituents are below this limit of easy visibility, are collected 
into another sub-class, which he terms residuary (Duplices 
reliquce). This arrangement is so far convenient, that after 
a little practice in the use of telescopes as applied to such 
objects, it is easy to judge what optical power will probably 
suffice to resolve a star of any proposed class and either sub- 
class, or would at least be so if the second or residuary sub- 
class were further subdivided by placing in a third sub-class 
"delicate" double stars, or those in which the companion 
star is so very minute as to require a high degree of optical 
power to perceive it, of which instances will presently be 
given. 



OUTLINES OF ASTRONOMY 



74S 



(836.) The following may be taken as specimens of each 
class. They are all taken from among the lucid, or con- 
spicuous stars, and to such of our readers as may be in pos- 
session of telescopes, and may be disposed to try them on 
such objects, will afford a ready test of their degree of 

efficiency. 

Class I., 0" to 1". 



y Coronae Bor. 


i) Coronae. 


1 Ophiuchi. 


Atlas Pleiadum. 


y Centauri. 


7) Herculis. 


<f> Draconis. 


4 Aquarii. 


y Lupi. 


A Cassiopeise. 


4> Ursae Majoris. 


5 Aquarii. 


c Arietis. 


A Ophiuchi. 


X Aquilaa. 


42 Comae. 


£ Herculis. 


n Lupi. 


w Leonis. 


52 Arietis. 


y 2 Andromedae. 


A Cygni. 


4> Andromedae. 


66 Piscium 




Class II., 


1" TO 2". 




y Circini. 


£, Boutis. 


£ Ursae Majoris. 


2 Camelopardi. 




t Cassiopeiae. 


ir Aquilae. 


32 Orionis. 


e Chamaeleontis. 


i 2 Cancri. 


<r Corouae Bor. 


52 Orionis. 




Class ILL, 


, 2" to 4". 




a Piscium. 


y Yh'ginis. 


£ Aquarii. 


M Draconis. 


Hydrae. 


S Serpentis. 


£ Orionis. 


ju. Canis. 


y Ceti. 


e Bootis. 


t Leonis. 


p Herculis. 


y Leonis. 


e Draconis. 


t Tiianguli. 


g- Cassiopeiae. 


y Coronae Aus. 


e Hydrae. 


k Leooris. 


44 Bootis. 




Class IV., 


4" to 8". 




a Crucis. 


Phcenicis. 


| Cephei. 


fi Eridani. 


a Herculis. 


k Cephei. 


w Bootis. 


70 Ophiuchi. 


a Gerninorum. 


A Orionis. 


p Capricorni. 


12 Eridani. 


8 Geminorum. 


M Oygai. 


v Argus. 


32 Eridani. 


£ Coronae Bor. 


| Bootis. 


(a Aurigae. 


95 Herculis. 




Class V., 


8" TO 12". 




Orion is. 


i Antliae. 


t Orionis. 


y Arietis. 


t\ Cassiopeiae. 


f Eridani. 


y Delphini. 


6 Eridani. 


2 Canum Ven. 




Class VI., i 


12" TO 16". 




a Centauri. 


y Volantis. 


k Bootis. 


Cephei. 


13 Lupi 


i. 


8 Monocerotis. 


Scorpii. 


£ Ursae Major. 


61 Cygni. 




Class VII., 


16" TO 24". 




a Canum Ven. 


Serpentis. 


24 Comae. 


e Norm as. 


k Coronae Aus. 


41 Draconis. 


£ Piscium. 


X Tauri. 


61 Ophiuchi. 




Class VIII. : 


, 24" TO 32". 




5 Herculis. 


< Herculis. 


X Cygni. 


\ Lyrse. 


i|/ Draconis. 


23 Orionis. 



T50 OUTLINES OF ASTRONOMY 

(837.) Among the most remarkable triple, quadruple, or 
multiple stars (for such also occur), may be enumerated, 



y Andromedse. 


6 Orionis. 


$ Scorpii. 


e Lyras. 


ix Lupi. 


11 Monocerotis. 


7? Cancri. 


fi Bootis. 


12 Lyncis. 



Of these, y Andromedse, fx Bootis, and n Lupi appear in tele- 
scopes, even of considerable optical power, only as ordinary 
double stars; and it is only when excellent instruments 
are used that their smaller companions are subdivided and 
found to be, in fact, extremely close double stars, e Lyrae 
offers the remarkable combination of a double-double star. 
Viewed with a telescope of low power it appears as a coarse 
and easily divided double star, but on increasing the magnify- 
ing power, each individual is perceived to be beautifully 
and closely double, the one pair being about 2\% the other 




about 3" asunder. Each of the stars C Cancri, £ Scorpii, 11 
Monocerotis, and 12 Lyncis consists of a principal star, 
closely double, and a smaller and more distant attendant, 
while 6 Orionis presents the phenomenon of four brilliant 
principal stars, of the respective 4th, 6th, 7th, and 8th mag- 
nitudes, forming a trapezium, the longest diagonal of which 
is 21" -4, and accompanied by two excessively minute and 
very close companions (as in the above figure), to per- 
ceive both which is one of the severest tests which can be 
applied to a telescope. 



OUTLINES OF ASTRONOMY 751 

(838.) Of the "delicate" sub-class of double stars, or 
those consisting of very large and conspicuous principal 
stars, accompanied by very minute companions, the follow- 
ing specimens may suffice: 



a 2 Cancri. 


a Polaris. 


c Ursse Majoris. 


1 Bootis. 


a 2 Capricorni. 


a Scorpii. 


r Circini. 


<f> Virginis. 


a Iudi. 


Aquarii. 


k Geininorum. 


X Eridani. 


a Lyrse. 


■j Hydra. 


H Persei. 


16 Aurigae. 



(839.) To the amateur of astronomy the double stars 
offer a subject of very pleasing interest, as tests of the 
psrformance of his telescopes, and by reason of the finely 
contrasted colors which many of them exhibit, of which 
more hereafter. But it is the high degree of physical in- 
terest which attaches to them, which assigns them a con- 
spicuous place in modern astronomy, and justifies the 
minute attention and unwearied diligence bestowed on 
the measurement of their angles of position and distances, 
and the continual enlargement of our catalogues of them 
by the discovery of new ones. It was, as we have seen, 
under an impression that such combinations, if diligently 
observed, might afford a measure of parallax through the 
periodical variations it might be expected to produce in 
the relative situation of the small attendant star, that Sir 
W. Herschel was induced (between the years 1779 and 1784) 
to form the first extensive catalogues of them, under the 
scrutiny of higher magnifying powers than had ever pre- 
viously been applied to such purposes. In the pursuit of 
this object, the end to which it was instituted as a means 
was necessarily laid aside for a time, until the accumulation 
of more abundant materials should have afforded a choice of 
stars favorably circumstanced for systematic observation. 
Epochal measures, however, of each star, were secured, 
and, on resuming the subject, his attention was altogether 



752 OUTLINES OF ASTRONOMY 

diverted from the original object of the inquiry by phe- 
nomena of a very unexpected character, which at once 
engrossed his whole attention. Instead of finding, as he 
expected, that annual fluctuation to and fro of one star of 
a double star with respect to the other — that alternate annual 
increase and decrease of their distance and angle of position, 
which the parallax of the earth's annual motion would pro- 
duce — he observed, in many instances, a regular progressive 
change; in some cases bearing chiefly on their distance — 
in others on their position, and advancing steadily in one 
direction, so as clearly to indicate either a real motion of 
the stars themselves, or a general rectilinear motion of the 
sun and whole solar system, producing a parallax of a higher 
order than would arise from the earth's orbitual motion, and 
which might be called systematic parallax. 

(840.) Supposing the two stars, and also the sun, in mo- 
tion independently of each other, it is clear that for the 
interval of several years, these motions must be regarded 
as rectilinear and uniform. Hence, a very slight acquaint- 
ance with geometry will suffice to show that the apparent 
motion of one star of a double star, referred to the other as 
a centre, and mapped down, as it were, on a plane in which 
that other shall be taken for a fixed or zero point, can be 
no other than a right line. This, at least, must be the case 
if the stars be independent of each other; bat it will be 
otherwise if they have a physical connection, such as, for 
instance, real proximity and mutual gravitation would 
establish. In that case, they would describe orbits round 
each other, and round their common centre of gravity; and 
therefore the apparent path of either, referred to the other 
as fixed, instead of being a portion of a straight line, would 
be bent into a curve concave toward that other. The ob- 



OUTLINES OF ASTRONOMY 75'S 

served motions, however, were so slow, that many years' 
observation was required to ascertain this point; and it was 
not, therefore, until the year 1803, twenty-five years from 
the commencement of the inquiry, that anything like a posi- 
tive conclusion could be come to respecting the rectilinear 
or orbitual character of the observed changes of position. 

(8-il.) In that, and the subsequent year, it was distinctly 
announced by him, in two papers, which will be found in 
the Transactions of the Royal Society for those years," that 
there exist sidereal systems, composed of two stars revolv- 
ing about each other in regular orbits, and constituting 
what may be termed binary stars, to distinguish them from 
double stars generally so called, in which these physically 
connected stars are confounded, perhaps, with others only 
optically double, or casually juxtaposed in the heavens at 
different distances from the eye; whereas the individuals 
of a binary star are, of course, equidistant from the eye, 
or, at least, cannot differ more in distance than the semi- 
diameter of the orbit they describe about each other, which 
is quite insignificant compared with the immense distance 
between them and the earth. Between fifty and sixty in- 
stances of changes, to a greater or less amount, in the angles 
of position of double stars, are adduced in the memoirs 
above mentioned; many of which are too decided, and too 
regularly progressive, to allow of their nature being mis- 
conceived. In particular, among the more conspicuous stars 
— Castor, y Yirginis, £ Ursse, 70 Ophiuchi, a and yj Coronse, 
£ Bootis, 7) Cassiopeiae, y Leonis, C Herculis, d Cygni, y 
Bootis, s 4 and e 5 Lyrse, A Ophiuchi, y. Draconis, and 
C Aquarii, are enumerated as among the most remarkable 

11 The announcement was in fact made in 1802, but unaccompanied by the 
observations establishing the fact. 



754 OUTLINES OF ASTRONOMY 

instances of the observed motion; and to some of them even 
periodic times of revolution are assigned; approximative 
only, of course, and rather to be regarded as rough guesses 
than as results of any exact calculation, for which the data 
were at the time quite inadequate. For instance, the revo- 
lution of Castor is set down at 334 years, that of y Yirginis 
at 708, and that of y Leonis at 1200 years. 

(842.) Subsequent observation has fully confirmed these 
results. Of all the stars above named, there is not one 
which is not found to be fully entitled to be regarded as 
binary; and, in fact, this list comprises nearly all the most 
considerable visible in our latitudes which have yet been 
detected, though (as attention has been closely drawn to 
the subject, and observations have multiplied) it has, of 
late, received large accessions. Upward of a hundred double 
stars, certainly known to possess this character, were enu- 
merated by M. Madler in 1841, 12 and more are emerging 
into notice with every fresh mass of observations which 
come before the public. They require excellent telescopes 
for their effective observation, being for the most part so 
close as to necessitate the use of very high magnifiers (such 
as would be considered extremely powerful microscopes if 
employed to examine objects within our reach), to perceive 
an interval between the individuals which compose them. 

(843.) It may easily be supposed, that phenomena of this 
kind would not pass without attempts to connect them with 
dynamical theories. From their first discovery, they were 
naturally referred to the agency of some power, like that 
of gravitation, connecting the stars thus demonstrated to 
be in a state of circulation about each other; and the ex- 
tension of the Newtonian law of gravitation to these remote 

12 Dorpat Observations, vol. ix. 1840 and 1841. 



OUTLINES OF ASTRONOMY 755 

systems was a step so obvious, and so well warranted by 
our experience of its all-sufficient agency in our own, as to 
have been expressly or tacitly made by every one who 
has given the subject any share of his attention. We owe, 
however, the first distinct system of calculation, by which 
the elliptic elements of the orbit of a binary star could be 
deduced from observations of its angle of position and 
distance at different epochs, to M. Savary, who showed, 13 
that the motions of one of the most remarkable among 
them (I Ursse) were explicable, within the limits allowable 
for error of observation, on the supposition of an elliptic 
orbit described in the short period of 58J years. A different 
process of computation conducted Professor Encke 14 to an 
elliptic orbit for 70 Ophiuchi, described in a period of 
seventy-four years. M. Madler has especially signalized 
himself in this line of inquiry (see Table). Several orbits 
have also been calculated by Mr. Hind, Admiral Smyth, 
Mr. Jacob, Mr. Powell, M. Villarceau, Professors Winnecke 
and Klinkerfues; and the author of these pages has himself 
attempted to contribute his mite to these interesting investi- 
gations. 15 The following may be stated as the chief results 
which, have been obtained in this branch of astronomy: 16 

13 Connoi33. des Temps, 1830. M Berlin Ephera. 1832. 

15 Mem. R. Ast. Soc. vols. v. and xviii. 

16 ^he "position of the node" in col. 4 expresses the angle of position (see 
art. 204) of the line of intersection of the plane of the orbit, with the plane of 
the heavens on which it is seen projected. The "inclination" in col. 6 is the 
inclination of these two planes to one another. Col. 5 shows the angle actually 
included in the plane of the orbit, between the line of nodes (defined as above) 
and the line of apsides. The elements assigned in this table to <»> Leonis, 
£ Bootis and Castor must be considered as very doubtful. Some cause of per- 
turbation has been suspected to exist in the movements of p Ophiuchi. Mr. 
Jacob, comparing some old (and no doubt very rude) observations by Richaud 
and Feuille, in 1690 and 1709, draws a similar conclusion in the case of the sys- 
tem of a. Centauri. Comparing the more modern (and only reliable observations), 
however, this opinion seems hardly entitled to the confidence with which he 
insists on it. A very few years' additional observation will decide the question. 
This magnificent double star well merits the most careful and diligent attention 
of astronomers. 



756 



OUTLINES OF ASTRONOMY 



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OUTLINES OF ASTRONOMY 75? 

(844.) Of the stars in the above list, that which has been 
most assiduously watched, and has offered phenomena of 
the greatest interest, is y Yirginis. It is a star of the vulgar 
3d magnitude (3*08=Photom. 3494), and its component 
individuals are very nearly equal, and as it would seem in 
some slight degree variable, since, according to the obser- 
vations of M. Struve, the one is alternately a little greater 
and a little less than the other, and occasionally exactly 
equal to it. It has been known to consist of two stars since 
the beginning of the eighteenth century; the distance being- 
then between six and seven seconds, so that any tolerably 
good telescope would resolve it. When observed by Her- 
schel in 1780, it was 5" -66, and continued to decrease grad- 
ually and regularly till at length, in 1836, the two stars had 
approached so closely as to appear perfectly round and 
single under the highest magnifying power which could 
be applied to most excellent instruments — the great re- 
fractor at Pulkova alone, with a magnifying power of 1000, 
continuing to indicate by the wedge-shaped form of the 
disk of the star its composite nature. By estimating the 
ratio of its length to its breadth and measuring the former, 
M. Struve concludes that, at this epoch (1836-41), the dis- 
tance of the two stars, centre from centre, might be stated 
at 0"-22. From that time the star again opened, and is now 
again a perfectly easily separable star. This very remark- 
able diminution and subsequent increase of distance has 
been accompanied by a corresponding and equally remark- 
able increase and subsequent diminution of relative angular 
motion. Thus, in the year 1783 the apparent angular mo- 
tion hardly amounted to half a degree per annum, while 
in 1830 it had increased to 5°, in 1834 to 20°, in 1835 to 40°, 
and about the middle of 1836 to upward of 70° per annum, 



758 OUTLINES OF ASTRONOMY 

or at the rate of a degree in five days. This is in entire 
conformity with the principles of dynamics, which establish 
a necessary connection between the angular velocity and the 
distance, as well in the- apparent as in the real orbit of one 
body revolving about another under the influence of mutual 
attraction; the former varying inversely as the square of the 
latter, in both orbits, whatever be the curve described and 
whatever the law of the attractive force. It fortunately 
happens that Bradley, in 1718, had noticed and recorded 
in the margin of one of his observation books, the apparent 
direction of the line of junction of the two stars, as seen on 
the meridian in his transit telescope, viz. parallel to the 
line joining two conspicuous stars a and d of the same con- 
stellation, as seen by the naked eye. This note, rescued 
from oblivion by the late Professor Eigaud, has proved 
of singular service in the verification of the elements 
above assigned to the orbit, which represent the whole 
series of recorded observations that date up to the end 
of 1846 (comprising an angular movement of nearly nine- 
tenths of a complete circuit), both in angle and distance, 
with a degree of exactness fully equal to that of obser- 
vation itself No doubt can, therefore, remain as to the 
prevalence in this remote system of the Newtonian law of 
gravitation. 

(845.) The observations of £ Ursse Majoris are equally 
well represented by M. Madler's elements (4 c of our table), 
thus fully justifying the assumption of the Newtonian law 
as that which regulates the motions of their binary systems. 
And even should it be the case, as M. Madler appears to 
consider, that in one instance at least (that of p Ophiuchi), 
deviations from elliptic motion, too considerable to arise 
from mere error of observation, exist (a position we are by 



OUTLINES OF ASTRONOMY 759 

no means prepared to grant 17 ), we should rather be disposed 
to look for the cause of such deviations in perturbations 
arising (as Bessel has suggested) from the large or central 
star itself being actually a close and hitherto unrecognized 
double star than in any defect of generality in the New- 
tonian law. 

(846.) If the great length of the periods of some of these 
bodies be remarkable, the shortness of those of others is 
hardly less so. C Herculis has already completed two revo- 
lutions since the epoch of its first discovery, exhibiting in 
its course the extraordinary spectacle of a sidereal occupa- 
tion, the small star having twice been hidden behind or be- 
fore the large one. -q Coronas, C Oancri, £ Ursae and a Cen- 
tauri have each performed more than one entire circuit, and 
70 Ophiuchi and y Yirginis have accomplished by far the 
larger portion of one in angular motion. If any doubt, 
therefore, could remain as to the reality of their orbitual 
motions, or any idea of explaining them by mere parallactic 
changes, or by any other hypothesis than the agency of 
centripetal force, these facts must suffice for their complete 
dissipation. We have the same evidence, indeed, of their 
rotations about each other, that we have of those of Uranus 
and Neptune about the sun; and the correspondence be- 
tween their calculated and observed places in such very 
elongated ellipses, must be admitted to carry with it proof 

17 p Ophiuchi belongs to the class of very unequal double star3, the magni- 
tudes of the individuals being 4 and 7. Such stars present difficulties in the 
exact measurement of their angles of position which even yet continue to em- 
barrass the observer, though, owing to later improvements in the art of execut- 
ing such measurements, their influence is confined within much narrower limits 
than in the earlier history of the subject. In simply placing a fine single wire 
parallel to the line of junction of two such stars it is easily possible to commit 
an error of 3° or 4°. By placing them between two parallel thick wires such 
errors are in great measure obviated. [The elements by Schur, in our table, 
art. 843, represent with the exactness of observation itself the whole series 
of positions and distances observed from 1779 to 1866.] 



760 OUTLINES OF ASTRONOMY 

of the prevalence of the Newtonian law of gravity in their 
systems, of the very same nature and cogency as that of the 
calculated and observed places of comets round the central 
body of our own. 

(847.) But it is not with the revolutions of bodies of a 
planetary or cometary nature round a solar centre that we 
are now concerned; it is with that of sun round sun — each, 
perhaps, at least in some binary systems where the individu- 
als are very remote and their period of revolution very long, 
accompanied with its train of planets and their satellites, 
closely shrouded from our view by the splendor of their re- 
spective suns, and crowded into a space bearing hardly a 
greater proportion to the enormous interval which separates 
them, than the distances of the satellites of our planets from 
their primaries bear to their distances from the sun itself. 
A less distinctly characterized subordination would be in- 
compatible with the stability of their systems, and with the 
planetary nature of their orbits. Unless closely nestled 
■under the protecting wing of their immediate superior, the 
sweep of their other sun in its perihelion passage round their 
own might carry them off, or whirl them into orbits utterly 
incompatible with the conditions necessary for the existence 
of their inhabitants. It must be confessed, that we have 
here a strangely wide and novel field for speculative excur- 
sions, and one which it is not easy to avoid luxuriating in. 

(848.) The discovery of the parallaxes of a Centauri and 
61 Oygni, both which are above enumerated among the 
" conspicuous" double stars of the 6th class (a distinction 
fully merited in the case of the former by the brilliancy of 
both its constituents), enables us to speak with an approach 
to certainty as to the absolute dimensions of both their 
orbits, and thence to form a probable opinion as to the 



OUTLINES OF ASTRONOMY 761 

general scale on which these astonishing systems are con- 
structed. The distance of the two stars of 61 Cygni sub- 
tends at the earth an angle which, since the earliest micro- 
metrical measures in 1781, has varied hardly 1" either way 
from a mean value 16" -5. On the other hand, the angle of 
position has altered since the same epoch by nearly 50°, so 
that it would appear probable that the true form of the orbit 
is not far from circular, its situation at right angles to the 
visual line, and its periodic time probably not short of 500 
years. Now, as the ascertained parallax of this star is 
0"'348, which is, therefore, the angle the radius of the 
earth's orbit would subtend if equally remote, it follows 
that the mean distance between the stars is to that radius, 
as 16" -5 : 0"-348, or as 4741 : 1. The orbit described by 
these two stars about each other undoubtedly, therefore, 
greatly exceeds in dimensions that described by Neptune 
about the sun. Moreover, supposing the period to be five 
centuries (and the distance being actually on the increase, 
it can hardly be less) the general propositions laid down by 
Newton, 18 taken in conjunction with Kepler's third law, en- 
able us to calculate the sum of the masses of the two stars, 
which, on these data, we find to be 0-353, the mass of our 
sun being 1. The sun, therefore, is neither vastly greater 
nor vastly less than the stars composing 61 Cygni. 

(849.) The data in the case of a Centauri are more uncer- 
tain. Since the year 1822, the distance has been steadily 
and pretty rapidly decreasing at an average rate of about 
half a second per annum, and that with little change till 
lately in the angle of position." Hence, it follows evidently 



18 Principia, 1. i. Props. 57, 58, 59. 

19 In the 70 years between Lacaille's observations and 1822, there exists no 
record of any observed angle of position. 



762 OUTLINES OF ASTRONOMY 

that the plane of its orbit passes nearly through the earth, 
and (the distance about the middle of 1834 having been 17J") 
it is very probable that either an occultation, like that ob- 
served in C Herculis, or a close appulse of the two stars, will 
take place about the year 1859. As the observations we 
possess afford no sufficient grounds for a satisfactory calcu- 
lation of elliptic elements we must be content to assume 
what, at all events, they fully justify, viz. that the major 
semiaxis must exceed 12", and is very probably consider- 
ably greater. Now this with a parallax of 0"-913 would give 
for the real value of the semiaxis 13-15 radii of the earth's 
orbit, as a minimum. The real dimensions of their ellipse, 
therefore, cannot be so small as the orbit of Saturn ; in all 
probability exceed that of Uranus; and may possibly be 
much greater than either. 

(850.) The parallel between these two double stars is a 
remarkable one. Owing no doubt to their comparative prox- 
imity to our system, their apparent proper motions are both 
unusually great, and for the same reason probably rather 
than owing to unusually large dimensions, their orbits ap- 
pear to us under what, for binary double stars, we must call 
unusually large angles. Each consists, moreover, of stars, 
not very unequal in brightness, and in each both the stars 
are of a high yellow approaching to orange color, the smaller 
individual, in each case, being also of a deeper tint. What- 
ever the diversity, therefore, which may obtain among other 
sidereal objects, these would appear to belong to the same 
family or genus. 20 

20 Similar combinations are very numerous. Many remarkable instances 
occur among the double stars catalogued by the author in the 2d, 3d, 4th, 6th 
and 9th volumes of Trans. Roy. Ast. Soc. and in the volume of Southern obser- 
vations already cited. See Nos. 121, 375, 1066, 1907, 2030, 2146, 2244, 2772, 
3853, 3935, 3998, 4000, 4055, 4196, 4210, 4615, 4649, 4765, 5003, 5012, of 



OUTLINES OF ASTRONOMY 763 

(851.) Many of the double stars exhibit the curious and 
beautiful phenomenon of contrasted or complementary col- 
ors. 21 In such instances, the larger star is usually of a 
ruddy or orange hue, while the smaller one appears blue 
or green, probably in virtue of that general law of optics, 
which provides, that when the retina is under the influence 
of excitement by any bright, colored light, feebler lights, 
which seen alone would produce no sensation but of white- 
ness, shall for the time appear colored with the tint comple- 
mentary to that of the brighter. Thus a yellow color pre- 
dominating in the light of the brighter star, that of the less 
bright one in the same field of view will appear blue; while, 
if the tint of the brighter star verge to crimson, that of the 
other will exhibit a tendency to green — or even appear as a 
vivid green, under favorable circumstances. The former 
contrast is beautifully exhibited by t Cancri — the latter by 
y Andromedae, 22 both fine double stars. If, however, the 
colored star be much the less bright of the two, it will not 
materially affect the other. Thus, for instance, ^ Cassiopeiae 
exhibits the beautiful combination of a large white star, and 
a small one of a rich ruddy purple. It is by no means, how- 
ever, intended to say, that in all such cases one of the colors 
is a mere effect of contrast, and it may be easier suggested 
in words, than conceived in imagination, what variety of 
illumination two suns — a red and a green, or a yellow and a 

these catalogues. The fine binary star, B. A. C. No. 4923, has its constituents 
15" apart, the one 6m. yellow, the other 7m. orange. 

81 " other suns, perhaps, 

"With their attendant moons thou wilt descry, 
Communicating male s,nd female light 
(Which two great sexes animate the world), 
Stored in each orb, perhaps, with some that live." 

— Paradise Lost, viii. 148. 
29 The small star of v Andromedae i3 close double. Both its individuals ar-} 
green: a similar combination, with even more decided colors, is presented tv' 
the double star, h. 881. 



764 



OUTLINES OF ASTRONOMY 



blue one— must afford a planet circulating about either; and 
what charming contrasts and "grateful vicissitudes" — a red 
and a green day, for instance, alternating with a white one 
and with darkness — might arise from the presence or ab- 
sence of one or other, or both, above the horizon. Insu- 
lated stars of a red color, almost as deep as that of blood, 28 
occur in many parts of the heavens, but no green or blue star 
(of any decided hue) has, we believe, ever been noticed un- 
associated with a companion brighter than itself. Many of 
the red stars are variable. 

(852.) Another very interesting subject of inquiry, in 
the physical history of the stars, is their proper motion. It 
was first noticed by Halley, that three principal stars, Sinus, 
Arcturus, and Aldebaran, are placed by Ptolemy, on the 
strength of observations made by Hipparchus, 130 years 
B.C., in latitudes respectively 20', 22', and 33' more north- 
erly than he actually found them in 1717. 24 Making due 
allowance for the diminution of obliquity of the ecliptic in 
the interval (1847 years) they ought to have stood, if really 
fixed, respectively 10', 14', and 0' more southerly. As the 
circumstances of the statement exclude the supposition of 
error of transcription in the MSS., we are necessitated to 



23 rp^ following are the R. ascensions and N. P. distances for 1830, of some 
of the most remarkable of these sanguine or ruby stars: 



R. A. 

h. m. s. 
4 40 53 

4 51 51 

5 38 29 
9 27 56 
9 48 31 



N. P. D. 



61 46 21 

105 2 4 

136 32 15 

152 2 48 

130 47 12 



R. A. 

h. m, s. 
10 52 10 
12 37 31 
16 29 44 

20 7 8 

21 15 36 



NP. D. 



107 24 40 

148 45 47 

2 

50 11 



122 
111 



48 22 



R. A. 

h, m, s. 
21 37 18 
21 37 20 
21 15 37 



N. P. D. 



31 59 47 
52 54 47 
48 8 12 



Of these No. 6 (in order of right ascension) is in the same field of view with 
o. Hydras et Crateris, and No. 7, with £ Crucis. No. 2 (in the same order) is 
variable. 

24 Phil. Trans. 1717, vol. xxx. fol. 736. 



OUTLINES OF ASTRONOMY 765 

admit a southward motion in latitude in these stars to the 
very considerable extent, respectively, of 37', 42', and 33', 
and this is corroborated by an observation of Aldebaran at 
Athens, in the year A.D. 509, which star, on the 11th of 
March in that year, was seen immediately after its emer- 
gence from occultation by the moon, in such a position as it 
could not have had if the occultation were not nearly cen- 
tral. Now, from the knowledge we have of the lunar mo- 
tions, this could not have been the case had Aldebaran at 
that time so much southern latitude as at present. A priori, 
it might be expected that apparent motions of some kind or 
other should be detected among so great a multitude of indi- 
viduals scattered through space, and with nothing to keep 
them fixed. Their mutual attractions even, however incon- 
ceivably enfeebled by distance, and counteracted by oppos- 
ing attractions from opposite quarters, must in the lapse of 
countless ages produce some movements — some change of in- 
ternal arrangement — resulting from the difference of the op- 
posing actions. And it is a fact, that such apparent motions 
are really proved to exist by the exact observations of mod- 
ern astronomy. Thus, as we have seen, the two stars of 61 
Cygni have remained constantly at the same, or very nearly 
the same, distance, of 15*, for at least fifty years past, al- 
though they have shifted their local situation in the heav- 
ens, in this interval of time, through no less than 4' 23", the 
annual proper motion of each star being 5" -3; by which 
quantity (exceeding a third of their interval) this system is 
every year carried bodily along in some unknown path, by a 
motion which, for many centuries, must be regarded as uni- 
form and rectilinear. Among stars not double, and no way 
differing from the rest in any other obvious particular, e Incli, 25 



25 D'Arrest. Astr. Nachr., No. 618; Argelander Do. No. 475. 



766 OUTLINES OF ASTRONOMY 

Grroomb. 1830, and /x Cassiopeiae are to be remarked as 
having the greatest proper motions of any yet ascer- 
tained, amounting respectively to 1" *74, 7 7/, 75 and 3"*74 of 
annual displacement. And a great many others have been 
observed to be thus constantly carried away from their 
places by smaller, but not less unequivocal motions. 26 

(853.) Motions which require whole centuries to accumu- 
late before they produce changes of arrangement, such as 
the naked eye can detect, though quite sufficient to destroy 
that idea of mathematical fixity which precludes specula- 
tion, are yet too trifling, as far as practical applications go, 
to induce a change of language, and lead us to speak of the 
stars in common parlance as otherwise than fixed. Small 
as they are, however, astronomers, once assured of their 
reality, have not been wanting in attempts to explain and 
reduce them to general laws. JTo one, who reflects with due 
attention on the subject, will be inclined to deny the high 
probability, nay certainty, that the sun as well as the stars 
must have a proper motion in some direction; and the in- 
evitable consequence of such a motion, if unparticipated by 
the rest, must be a slow average apparent tendency of all the 
stars to the vanishing point of lines parallel to that direc- 
tion, and to the region which he is leaving, however greatly 
individual stars might differ from such average by reason of 
their own peculiar proper motion. This is the necessary 
effect of perspective; and it is certain that it must be de- 
tected by observation, if we knew accurately the apparent 
proper motions of all the stars, and if we were sure that they 
were independent, i.e. that the whole firmament, or at least 



26 The reader may consult "a list of 314 stars having, or supposed to have, 
a proper motion of not less than about 0"'5 of a great circle" (per annum) by 
the late F. Baily, Esq. Trans. Ast. Soc. v. p. 158. 



OUTLIXES OF ASTRONOMY 767 

all that part which we see in our own neighborhood, were 
not drifting along together, by a general set as it were, in 
one direction, the result of unknown processes and slow in- 
ternal changes going on in the sidereal stratum to which our 
system belongs, as we see motes sailing in a current of air, 
and keeping nearly the same relative situation with respect 
to one another. 

(854:.) It was on this assumption, tacitly made indeed, 
but necessarily implied in every step of his reasoning, that 
Sir William Herschel, in 1783, on a consideration of the 
apparent proper motions of such stars as could at that period 
be considered as tolerably (though still imperfectly) ascer- 
tained, arrived at the conclusion that a relative motion of 
the sun, among the fixed stars in the direction of a point or 
parallactic apex, situated near X Herculis, that is to say, 
in E. A. 17* 22 m =260° 34', N. P. D. 63° 43' (1790), would 
account for the chief observed apparent motions, leaving, 
however, some still outstanding and not explicable by this 
cause; and in the same year Prevost, taking nearly the same 
view of the subject, arrived at a conclusion as to the solar 
apex (or point of the sphere toward which the sun relatively 
advances), agreeing nearly in polar distance with the fore- 
going, but differing from it about 27° in right ascension. 
Since that time methods of calculation have been improved 
and concinnated, our knowledge of the proper motions of 
the stars has been rendered more precise, and a greater num- 
ber of cases of such motions have been recorded. The sub- 
ject has been resumed by several eminent astronomers and 
mathematicians: viz. 1st, by M. Argelander, who, from the 
consideration of the proper motions of 21 stars exceeding 1" 
per annum in arc, has placed the solar apex in R. A. 256° 
25', K. P. D. 51° 23'; from those of 50 stars between 0"-5 



76& OUTLINES OF ASTRONOMY 

and r-0, in 255° 10', 51° 26'; and from those of 319 stars 
having motions between 0"*1 and 0^*5 per annum, in 261° 11', 
59° 2'; 2dly, by M. Luhndahl, whose calculations, founded 
on the proper motions of 147 stars, give 252° 53', 75° 34'; 
and, 3dly, by M. Otto Struve, whose result, 261° 22', 62° 24', 
emerges from a very elaborate discussion of the proper mo- 
tions of 392 stars. All these places are for A.D. 1790. 

(855.) The most probable mean of the results obtained 
by these three astronomers is (for the same epoch) E. A. = 
259° 9', N. P. D. 55° 23'. Their researches, however, ex- 
tending only to stars visible in European observatories, 
it became a point of high interest to ascertain how far the 
stars of the southern hemisphere not so visible, treated inde- 
pendently on the same system of procedure, would corrobo- 
rate or controvert their conclusion. The observations of 
Lacaille, at the Cape of Good Hope, in 1751 and 1752, 
compared with those of Mr. Johnson at St. Helena, in 
1829-33, and of Henderson at the Cape in 1830 and 1831, 
have afforded the means of deciding this question. The 
task has been executed in a masterly manner by Mr. Gal- 
loway, in a paper published in the Philosophical Transac- 
tions for 1841 (to which we may also refer the reader for 
a more particular account of the history of the subject than 
our limits allow us to give). On comparing the records, 
Mr. Galloway finds eighty- one southern stars not employed 
in the previous investigations above referred to, whose 
proper motions in the intervals elapsed appear consider- 
able enough to assure us that they have not originated in 
error of the earlier observations. Subjecting these to the 
same process of computation he concludes for the place 
of the solar apex, for 1790, as follows: viz. E. A. 260° 
1', N. P. D. 55°, 37', a result so nearly identical with that 



OUTLINES OF ASTRONOMY 769 

afforded by the northern hemisphere, as to afford a full 
conviction of its near approach to truth, and what may 
fairly be considered a demonstration of the physical cause 
assigned. 

(856.) Of the mathematical conduct of this inquiry the 
nature of this work precludes our giving any account; but 
as the philosophical principle on which it is based has been 
misconceived, it is necessary to say a few words in explana- 
tion of it. Almost all the greatest discoveries in astronomy 
have resulted from the consideration of what we have else- 
where termed residual phenomena, 27 of a quantitative or 
numerical kind, that is to say, of such portions of the 
numerical or quantitative results of observation as remain 
outstanding and unaccounted for after subducting and 
allowing for all that would result from the strict applica- 
tion of known principles. It was thus that the grand dis- 
covery of the precession of the equinoxes resulted as a 
residual phenomenon, from the imperfect explanation of 
the return of the seasons by the return of the sun to the 
same 'apparent place among the fixed stars. Thus, also, 
aberration and nutation resulted as residual phenomena 
from that portion of the changes of the apparent places 
of the fixed stars which was left unaccounted for by pre- 
cession. And thus again the apparent proper motions of 
the stars are the observed residues of their apparent move- 
ments outstanding and unaccounted for by strict calculation 
of the effects of precession, nutation, and aberration. The 
nearest approach which human theories can make to perfec- 
tion is to diminish this residue, this caput mortuum of obser- 
vation, as it may be considered, as much as practicable, 

27 Discourse on the Study of Natural Philosophy (1833). Cab. Cyclopaedia, 
No. 14. 



770 OUTLINES OF ASTRONOMY 

and, if possible, to reduce it to nothing, either by showing 
that something has been neglected in our estimation of 
known causes, or by reasoning upon it as a new fact, and 
on the principle of the inductive philosophy ascending 
from the effect to its cause or causes. On the suggestion 
of any new cause hitherto unresorted to for its explanation, 
our first object must of course be to decide whether such 
a cause would produce such a result in kind: the next, to 
assign to it such an intensity as shall account for the 
greatest possible amount of the residual matter in hand. 
The proper motion of the sun being suggested as such a 
cause, we have two things disposable — its direction and 
velocity, both which it is evident, if they ever become 
known to us at all, can only be so by the consideration of 
the very phenomenon in question. Our object, of course, 
is to account, if possible, for the whole of the observed 
proper motions by the proper assumption of these elements. 
If this be impracticable, what remains unaccounted for is 
a residue of a more recondite kind, but which, so long as 
it is unaccounted for, we must regard as purely casual, 
seeing that, for anything we can perceive to the contrary, 
it might with equal probability be one way as the other. 
The theory of chances, therefore, necessitates (as it does 
in all such cases) the application of a general mathematical 
process, known as "the method of least squares," which 
leads, as a matter of strict geometrical conclusion, to the 
values of the elements sought, which, under all the circum- 
stances, are the most probable. 

(857.) This is the process resorted to by all the geometers 
we have enumerated in the foregoing articles (arts. 854, 855). 
It gives not only the direction in space, but also the velocity 
of the solar motion, estimated on a scale conformable to that 



OUTLINES OF ASTR0N03IY 771 

in which the velocity of the sidereal motions to be explained 
are given; i.e. in seconds of arc as subtended at the average 
distance of the stars concerned, by its annual motion in 
space. But here a consideration occurs which tends materi- 
ally to complicate the problem, and to introduce into its 
solution an element depending on suppositions more or 
less arbitrary. The distance of the stars being, except 
in two or three instances, unknown, we are compelled 
either to restrict our inquiry to these, which are too few 
to ground any result on, or to make some supposition as 
to the relative distances of the several stars employed. 
In this we have nothing but general probability to guide 
us, and two courses only present themselves, either, 1st, 
To class the distances of the stars according to their mag- 
nitudes, or apparent brightnesses, and to institute separate 
and independent calculations for each class, including stars 
assumed to be equidistant, or nearly so: or, 2dly, To class 
them according to the observed amount of their apparent 
proper motions, on the presumption that those which appear 
to move fastest are really nearest to us. The former is the 
course pursued by M. Ofcto Struve, the latter by M. Ar- 
gelander. With regard to this latter principle of classifica- 
tion, however, two considerations interfere with its appli- 
cability, viz. 1st, that we see the real motion of the stars 
foreshortened by the effect of perspective; and 2dly, that 
that portion of the total apparent proper motion which 
arises from the real motion of the sun depends, not simply 
on the absolute distance of the star from the sun, but also 
on its angular apparent distance from the solar apex, being, 
coeteris paribus, as the sine of that angle. To execute such 
a classification correctly, therefore, we ought to know both 

these particulars for each star. The first is evidently out 

Astronomy— Vol. XX— 14 



772 OUTLINES OF ASTRONOMY 

of our reach. We are, therefore, for that very reason, com- 
pelled to regard it as casual, and to assume that on the 
average of a great number of stars it would be uninfluential 
on the result. But the second cannot be so summarily dis- 
posed of. By the aid of an approximate knowledge of the 
solar apex, it is true, approximate values may be found of 
the simply apparent portions of the proper motions, sup- 
posing all the stars equidistant, and these being subducted 
from the total observed motions, the residues might afford 
ground for the classification in question. 28 This, however, 
would be a long, and to a certain extent precarious system 
of procedure. On the other hand, the classification by ap- 
parent brightness is open to no such difficulties, since we 
are fully justified in assuming that, on a general average, 
the brighter stars are the nearer, and that the exceptions 
to this rule are casual in that sense of the word which it 
always bears in such inquiries, expressing solely our igno- 
rance of any ground for assuming a bias one way or other 
on either side of a determinate numerical rule. In Mr. Gral- 
oway's discussion of the southern stars the consideration 
of distance is waived altogether, which is equivalent to an 
admission of complete ignorance on this point, as well 
as respecting the real directions and velocities of the 
individual motions. 

(858.) The velocity of the solar motion which results 
from M. Otto Struve's calculations is such as would carry 
it over an angular subtense of //, 3392 if seen at right angles 
from the average distance of a star of the first magnitude. 

28 M. Argelander's classes, however, are constructed without reference to 
this consideration, on the sole basis of the total apparent amount of proper 
motion, and are, therefore, pro tanto, questionable. It is the more satisfactory- 
then to find so considerable an agreement among his partial results as actually 

obtains. 



OUTLINES OF ASTRONOMY 773 

li we take, with M. Struve, senior, the parallax of such 
a star as probably equal to 0"-209, a9 we shall at once be 
enabled to compare this annual motion with the radius 
of the earth's orbit, the result being 1*623 of such units. 
The sun then advances through space (relatively, at least, 
among the stars), carrying with it the whole planetary and 
cometary system with a velocity of 1-623 radii of the earth's 
orbit, so or 154,185,000 miles per annum, or 422,000 miles 
(that is to say, nearly its own semidiameter) per diem: in 
other words, with a velocity a very little greater than one- 
fourth of the earth's annual motion in its orbit. 

(859.) Another generation of astronomers, perhaps many, 
must pass away before we are in a condition to decide from 
a more precise and extensive knowledge of the proper mo- 
tions of the stars than we at present possess, how far the 
direction and velocity above assigned to the solar motion 
deviates from exactness, whether it continue uniform, and 
whether it show any sign of deflection from rectilinearity; 
so as to hold out a prospect of one day being enabled to 
trace out an arc of the solar orbit, and to indicate the direc- 
tion in which the preponderant gravitation of the sidereal 
firmament is urging the central body of our system. An 
analogy for such deviation from uniformity would seem 
to present itself in the alleged existence of a similar devia- 
tion in the proper motions of Sirius and Procyon, both 
which stars are considered to have varied sensibly in this 
respect within the limits of authentic and dependable ob- 
servation. Such, indeed, would appear to be the amount 
of evidence for this as a matter oifact as to have given rise 



29 Etudes d'Astronoinie Stellaire, p. lOfr. 

30 Mr. Airy (Mem. Ast. Soc. xxviii.) makes this velocity materially greater. 
See. however, Note L. 



774 OUTLINES OF ASTRONOMY 

to a speculation on the probable circulation of these stars 
round opaque (and therefore invisible) bodies at no great 
distances from them respectively, in the manner of binary- 
stars: [and it has been recently shown by M. Peters (Ast. 
Nachr. 748) that, in the case of Sirius, such a circulation, 
performed in a period of 50*093 years in an ellipse whose 
excentricity is 0-7994, the perihelion passage taking place 
at the epoch A.D. 1791-431, would reconcile the observed 
anomalies, and reduce the residual motion to uniformity. 
See Note J.] 

(860.) The whole of the reasoning upon which the de- 
termination of the solar motion in space rests, is based 
upon the entire exclusion of any laio either derived from 
observation or assumed in theory, affecting the amount and 
direction of the real motions both of the sun and stars. 
It supposes an absolute non-recognition, in those motions, 
of any general directive cause, such as, for example, a 
common circulation of all about a common centre. Any 
such limitation introduced into the conditions of the prob- 
lem of the solar motion would alter in toto both its nature 
and the form of its solution. Suppose for instance that, 
conformably to the speculations of several astronomers, 
the whole system of the Milky Way, including our sun, 
and the stars, our more immediate neighbors, which consti- 
tute our sidereal firmament, should have a general move- 
ment of rotation in the plane of the galactic circle (any 
other would be exceedingly improbable, indeed hardly 
reconcilable with dynamical principles), being held to- 
gether in opposition to the centrifugal force thus generated 
by the mutual gravitation of its constituent stars. Except 
we at the same time admitted that the scale on which this 
movement proceeds is so enormous that all the stars whose 



OUTLINES OF ASTRONOMY 775 

proper motions we include in our calculations go together 
in a body, so far as that movement is concerned (as forming 
too small an integrant portion of the whole to differ sensibly 
in their relation to its central point); we stand precluded 
from drawing any conclusion whatever, not only respecting 
the absolute motion of the sun, but respecting even its rela- 
tive movement among those stars, until we have established 
some law, or at all events framed some hypothesis having 
the provisional force of a law, connecting the whole, or 
a part of the motion of each individual with its situation 
in space. 

(861.) Speculations of this kind have not been wanting 
in astronomy, and recently an attempt has been made by 
M. Madler to assign the local centre in space, round which 
the sun and stars revolve, which he places in the group of 
the Pleiades, a situation in itself utterly improbable, lying 
as it does no less than 26° out of the plane of the galactic 
circle, out of which plane it is almost inconceivable that any 
general circulation can take place. In the present defective 
state of our knowledge respecting the proper motion of the 
smaller stars, especially in right ascension (an element for 
the most part far less exactly ascertainable than the polar 
distance, or at least which has been hitherto far less accu- 
rately ascertained), we cannot but regard all attempts of the 
kind as to a certain extent premature, though by no means 
to be discouraged as forerunners of something more deci- 
sive. The question, as a matter of fact, whether a rotation 
of the galaxy in its own plane exist or not might be at once 
resolved by the assiduous observation both in K. A. and 
polar distance of a considerable number of stars of the 
Milky Way, judiciously selected for the purpose, and in- 
cluding all magnitudes, down to the smallest distinctly iden- 



776 OUTLINES OF ASTRONOMY 

tifiable, and capable of being observed with normal accu- 
racy: and we would recommend the inquiry to the special 
attention of directors of permanent observatories, provided 
with adequate instrumental means, in both hemispheres. 
Thirty or forty years of observation perseveringly directed 
to the object in view, could not fail to settle the question. 31 

(862.) The solar motion through space, if real and not 
simply relative, must give rise to uranographical corrections 
analogous to parallax and aberration. The solar or system- 
atic parallax is no other than that part of the proper motion 
of each star which is simply apparent, arising from the sun's 
motion, and until the distances of the stars be known, must 
remain inextricably mixed up with the other or real portion. 
The systematic aberration, amounting at its maximum (for 
stars 90° from the solar apex) to about 5", displaces all the 
stars in great circles diverging from that apex through an- 
gles proportional to the sines of their respective distances 
from it. This displacement, however, is permanent, and 
therefore uncognizable by any phenomenon, so long as the 
solar motion remains invariable; but should it, in the course 
of ages, alter its direction and velocity, both the direction 
and amount of the displacement in question would alter with 
it. The change, however, would become mixed up with 
other changes in the apparent proper motions of the stars, 
and it would seem hopeless to attempt disentangling them. 

(863.) A singular, and at first sight paradoxical effect of 
the progressive movement of light, combined with the 



31 An examination of the proper motions of the stars of the B. Assoc. Catal. 
in the portion of the Milky Way nearest either pole (where the motion should 
be almost wholly in R. A.) indicates no distinct symptom of such a rotation. If 
the question be taken up fundamentally, it will involve a redetermination from 
the recorded proper motions, both of the precession of the equinoxes and the 
change of obliquity of the ecliptic. 



OUTLINES OF ASTRONOMY 777 

proper motion of the stars, is, that it alters the apparent 
periodic time in which the individuals of a binary star cir- 
culate about each other. 33 To make this apparent, suppose 
them to circulate round each other in a plane perpendicular 
to the visual ray in a period of 10,000 days. Then if both 
the sun and the centre of gravity of the binary system re- 
mained fixed in space, the relative apparent situation of the 
stars would be exactly restored to its former state after the 
lapse of this interval, and if the angle of position were 0° at 
first, after 10,000 days it would again be so. But now sup- 
pose that the centre of gravity of the star were in the act of 
receding in a direct line from the sun with a velocity of one- 
tenth part of the radius of the earth's orbit per diem. Then 
at the expiration of 10,000 days it would be more remote 
from us by 1000 such radii, a space which light would re- 
quire 5*7 days to traverse. Although really, therefore, the 
stars would have arrived at the position 0° at the exact ex- 
piration of 10,000 days, it would require 5*7 days more for 
the notice of that fact to reach our system. In other words, 
the period would appear to us to be 10, 005*7 clays, since we 
could only conclude the period to be completed when to 
us as observers the original angle of position was again re- 
stored. A contrary motion would produce a contrary effect. 

32 Astronomische Nackrichlen, No. 520, by the Author. 



778 OUTLINES OF ASTRONOMY 



CHAPTER XVII 

OF CLUSTERS OF STARS AND NEBULA 

Of Clustering Groups of Stars — Globular Clusters — Their Stability Dynami- 
cally Possible — List of the most Remarkable — Classification of Nebulae 
and Clusters — Their Distribution over the Heavens — Irregular Clusters 
— Resolvability of Nebulae — Theory of the Formation of Clusters by 
Nebulous Subsidence — Of Elliptic Nebulas — That of Andromeda — An- 
nular and Planetary Nebulae — Double Nebulae — Nebulous Stars — Con- 
nection of Nebulae with Double Stars — Insulated Nebulae of Forms 
not Wholly Irregular — Of Amorphous Nebulae — Their Law of Distri- 
bution Marks them as Outliers of the Galaxy — Nebulas and Nebulous 
Group of Orion — Of Argo — Of Sagittarius — Of Cygnus — The Magel- 
lanic Clouds — Singular Nebula in the Greater of Them — Variable 
Nebulae — The Zodiacal Light — Shooting Stars — Speculations on the 
Dynamical Origin of the Sun's Heat 

(864.) When we cast our eyes over the concave of the 
heavens in a clear night, we do not fail to observe that here 
and there are groups of stars which seem to be compressed 
together in a more condensed manner than in the neighbor- 
ing parts, forming bright patches and clusters, which attract 
attention, as if they were there brought together by some 
general cause other than casual distribution. There is a 
group, called the Pleiades, in which six or seven stars may 
be noticed, if the eye be directed full upon it; and many more 
if the eye be turned carelessly aside while the attention is kept 
directed 1 upon the group. Telescopes show fifty or sixty 

1 It is a very remarkable fact that the centre of the visual area is far less 
sensible to feeble impressions of light, than the exterior portions of the retina. 
Few persons are aware of the extent to which this comparative insensibility 
extends, previous to trial. To estimate it, let the reader look alternately full at 
a star of the fifth magnitude, and beside it; or choose two, equally bright, and 
about 3° or 4° apart, and look full at one of them, the probability is, he will see 
only the other. The fact accounts for the multitude of stars with which we are 
impressed by a general view of the heavens ; their paucity when we come to 
count them. 



OUTLINES OF ASTRONOMY 779 

large stars thus crowded together in a very moderate space, 
comparatively insulated from the rest of the heavens. The 
constellation called Coma Berenices is another such group, 
more diffused, and consisting on the whole of larger stars. 
(865.) In the constellation Cancer, there is a somewhat 
similar, but less definite, luminous spot, called Prgesepe, or 
the bee-hive, which a very moderate telescope — an ordinary 
night-glass, for instance — resolves entirely into stars. In 
the sword-handle of Perseus, also, is another such spot, 
crowded with stars, which requires rather a better tele- 
scope to resolve into individuals separated from each other. 
These are called clusters of stars; and, whatever be their 
nature, it is certain that other laws of aggregation subsist in 
these spots, than those which have determined the scatter- 
ing of stars over the general surface of the sky. This con- 
clusion is still more strongly pressed upon us, when we 
come to bring very powerful telescopes to bear on these 
and similar spots. There are a great number of objects 
which have been mistaken for comets, and, in fact, have 
very much the appearance of comets without tails: small 
round, or oval nebulous specks, which telescopes of moder- 
ate power only show as such. Messier has given, in the 
Connois. cles Temj^s for 1784, a list of the places of 103 ob- 
jects of this sort; which all those who search for comets 
ought to be familiar with, to avoid being misled by their 
similarity of appearance. That they are not, however, com- 
ets, their fixity sufficiently proves; and when we come to 
examine them with instruments of great power- — such as re- 
flectors of eighteen inches, two feet, or more in aperture — 
any such idea is completely destroyed. They are then, for 
the most part, perceived to consist entirely of stars crowded 
together so as to occupy almost a definite outline, and to 



780 OUTLINES OF ASTRONOMY 

run up to a blaze of light in the centre, where their conden- 
sation is usually the greatest. (See fig. 1, Plate II., which 
represents (somewhat rudely) the thirteenth nebula of Mes- 
sier's list (described by him as nebuleuse sans Stoiles), as seen 
in a reflector of 18 inches aperture and 20 feet focal length.) 
Many of them, indeed, are of an exactly round figure, and 
convey the complete idea of a globular space filled full of 
stars, insulated in the heavens, and constituting in itself a 
family or society apart from the rest, and subject only to its 
own internal laws. It would be a vain task to attempt to 
count the stars in one of these globular clusters. They are 
not to be reckoned by hundreds; and on a rough calcula- 
tion, grounded on the apparent intervals between them at 
the borders, and the angular diameter of the whole group, 
it would appear that many clusters of this description must 
contain, at least, five thousand stars, compacted and wedged 
together in a round space, whose angular diameter does not 
exceed eight or ten minutes; that is to say, in an area not 
more than a tenth part of that covered by the moon. 

(866.) Perhaps it may be thought to savor of the gigan- 
tesque to look upon the individuals of such a group as suns 
like our own, and their mutual distances as equal to those 
which separate our sun from the nearest fixed star: yet, 
when we consider that their united lustre affects the eye 
with a less impression of light than a star of the fourth mag- 
nitude (for the largest of these clusters is barely visible to 
the naked eye), the idea we are thus compelled to form of 
their distance from us may prepare us for almost any esti- 
mate of their dimensions. At all events, we can hardly 
look upon a group thus insulated, thus in seipso totus, teres, 
atque rotundus, as. not forming a system of a peculiar and 
definite character. Their round figure clearly indicates the 



OUTLINES OF ASTRONOMY 781 

existence of some general bond of union in the nature of an 
attractive force; and, in many of them, there is an evident 
acceleration in the rate of condensation as we approach the 
centre, which is not referable to a merely uniform distribu- 
tion of equidistant stars through a globular space, but marks 
an intrinsic density in their state of aggregation, greater in 
the centre than at the surface of the mass. It is difficult 
to form any conception of the dynamical state of such a sys- 
tem. On the one hand, without a rotatory motion and a 
centrifugal force, it is hardly possible not to regard them as 
in a state of progressive collapse. On the other, granting 
such a motion and such a force, we find it no less difficult 
to reconcile the apparent sphericity of their form with a 
rotation of the whole system round any single axis, without 
which internal collisions might at first sight appear to be in- 
evitable. If we suppose a globular space filled with equal 
stars, uniformly dispersed through it, and very numerous, 
each of them attracting every other with a force inversely as 
the square of the distance, the resultant force by which any 
one of them (those at the surface alone excepted) will be 
urged, in virtue of their joint attractions, will be directed 
toward the common centre of the sphere, and will be di- 
rectly as the distance therefrom. This follows from what 
Newton has proved of the internal attraction of a homogene- 
ous sphere. (See also note on art. 735.) Now, under such 
a law of force, each particular star would describe a perfect 
ellipse about the common centre of gravity as its centre, and 
that, in whatever plane and whatever direction it might re- 
volve. The condition, therefore, of a rotation of the clus- 
ter, as a mass, about a single axis would be unnecessary. 
Each ellipse, whatever might be the proportion of its axis, 
or the inclination of its plane to the others, would be invari- 



782 



OUTLINES OF ASTRONOMY 



able in every particular, and all would be described in one 
common period, so that at the end of every such period, or 
annus magnus of the system, every star of the cluster (ex- 
cept the superficial ones) would be exactly re-established in 
its original position, thence to set out afresh, and run the 
same unvarying round for an indefinite succession of ages. 
Supposing their motions, therefore, to be so adjusted at any 
one moment as that the orbits should not intersect each 
other, and so that the magnitude of each star, and the sphere 
of its more intense attraction, should bear but a small pro- 
portion to the distance separating the individuals, such a 
system, it is obvious, might subsist, and realize, in great 
measure, that abstract and ideal harmony, which Newton, 
in the 89th Proposition of the First Book of the Principia y 
has shown to characterize a law of force directly as the dis- 
tance. 2 

(867.) The following are the places, for 1830, of the 
principal of these remarkable objects, as specimens of 
their class: 



R, A. 


N. P. 


D. 


R. A. 


N. P. 


D. 


R. A. 


N. P. 


D. 


h. 


m. 


s. 


o 


' 


h. 


m. 


s. 


o 


' 


h. 


m. 


s. 


o 


r 





16 


25 


163 


2 


15 


9 


56 


87 


16 


17 


26 


51 


143 


34 


9 


8 


33 


154 


10 


15 


34 


56 


127 


13 


17 


28 


42 


93 


8 


12 


47 


41 


159 


57 


16 


6 


55 


112 


33 


11 


26 


4 


114 


2 


13 


4 


30 


70 


55 


16 


23 


2 


102 


40 


18 


55 


49 


150 


14 


13 


16 


38 


136 


35 


16 


35 


37 


53 


13 


21 


21 


43 


78 


34 


13 


34 


10 


60 


46 


16 


50 


24 


119 


51 


21 


24 


40 


91 


34 



Of these, by far the most conspicuous and remarkable 
is to Centauri, the fifth of the list in order of right ascen- 
sion. It is visible to the naked eye as a dim round coinetic 
object about equal to a star 4*5m., though probably if con- 



See also Quarterly Review, No. 94, p. 540. 



OUTLINES OF ASTRONOMY 783 

centred in a single point, the impression on the eye would 
be much greater. Viewed in a powerful telescope it appears 
as a globe of fully 20' in diameter, very gradually increasing 
in brightness to the centre, and composed of innumerable 
stars of the 13th and 15th magnitudes (the former probably 
being two or more of the latter closely juxtaposed). The 
11th in order of the list (R. A. 16 h 35 m ) is also visible to the 
naked eye in very fine nights, between -q and C Herculis, and 
is a superb object in a large telescope. Both were discov- 
ered by Halley, the former in 1677 and the latter in 1714. 

(868.) It is to Sir William Herschel that we owe the 
most complete analysis of the great variety of those ob- 
jects which are generally classed under the common head 
of Nebulas, bat which have been separated by him into — 
1st. Clusters of stars, in which the stars are clearly distin- 
guishable; and these, again, into globular and irregular 
clusters; 2d. Resolvable nebulae, or such as excite a sus- 
picion that they consist of stars, and which any increase 
of the optical power of the telescope may be expected to 
resolve into distinct stars; 3d. Nebulas, properly so called, 
in which there is no appearance whatever of stars; which, 
again, have been subdivided into subordinate classes, ac- 
cording to their brightness and size; 4th. Planetary nebulas; 
5th. Stellar nebulas; and, 6th. Nebulous stars. The great 
power of his telescopes disclosed the existence of an im- 
mense number of these objects before unknown, and showed 
them to be distributed over the heavens, not by any means 
uniformly, but with a marked preference for a certain dis- 
trict, extending over the northern pole of the galactic circle, 
and occupying the constellations Leo, Leo Minor, the body, 
tail, and hind legs of Ursa Major, Canes Yenatici, Coma 
Berenices, the preceding leg of Bootes, and the head, wings, 



784 OUTLINES OF ASTRONOMY 

and shoulder of Yirgo. In this region, occupying about 
one-eighth of the whole surface of the sphere, one-third 
of the entire nebulous contents of the heavens are congre- 
gated. On the other hand, they are very sparingly scattered 
over the constellations Aries, Taurus, the head and shoul- 
ders of Orion, Auriga, Perseus, Camelopardalus, Draco, 
Hercules, the northern part of Serpentarius, the tail of 
Serpens, that of Aquila, and the whole of Lyra. The hours 
3, 4, 5, and 16, 17, 18, of right ascension in the northern 
hemisphere are singularly poor, and, on the other hand, 
the hours 10, 11, and 12 (but especially 12), extraordinarily 
rich in these objects. In the southern hemisphere a much 
greater uniformity of distribution prevails, and with excep- 
tion of two very remarkable centres of accumulation, called 
the Magellanic clouds (of which more presently), there is 
no very decided tendency to their assemblage in any par- 
ticular region. 

(869.) Clusters of stars are either globular, such as we 
have already described, or of irregular figure. These latter 
are, generally speaking, less rich in stars, and especially 
less condensed toward the centre. They are also less defi- 
nite in outline; so that it is often not easy to say where 
they terminate, or whether they are to be regarded other- 
wise than as merely richer parts of the heavens than those 
around them. Many, indeed the greater portion of them, 
are situated in or close on the borders of the Milky Way. 
In some of them the stars are nearly all of a size, in others 
extremely different; and it is no uncommon thing to find 
a very red star much brighter than the rest, occupying a 
conspicuous situation in them. Sir William Herschel re- 
gards these as globular clusters in a less advanced state of 
condensation, conceiving all such groups as approaching. 



OUTLINES OF ASTRONOMY 785 

by their mutual attraction, to the globular figure, and 
assembling themselves together from all the surrounding 
region, under laws of which we have, it is true, no other 
proof than the observance of a gradation by which their 
characters shade into one another, so that it is impossible 
to say where one species ends and the other begins. Among 
the most beautiful objects of this class is that which sur- 
rounds the star x Crucis, set down as a nebula by Lacaille. 
It occupies an area of about one 48th part of a square de- 
gree, and consists of about 110 stars from the 7th magnitude 
downward, eight of the more conspicuous of which are 
colored with various shades of red, green, and blue, so as 
to give to the whole the appearance of a rich piece of 
jewelry. 

(870.) Kesolvable nebulas can, of course, only be con- 
sidered as clusters either too remote, or consisting of stars 
intrinsically too faint to affect us by their individual light, 
unless where two or three happen to be close enough to 
make a joint impression, and give the idea of a point 
brighter than the rest. They are almost universally round 
or oval — their loose appendages, and irregularities of form, 
being as it were extinguished by the distance, and the only 
general figure of the more condensed parts being discernible. 
It is under the appearance of objects of this character that 
all the greater globular clusters exhibit themselves in tel- 
escopes of insufficient optical power to show them well; 
and the conclusion is obvious, that those which the most 
powerful can barely render resolvable, and even those 
which, with such powers as are usually applied, show no 
sign of being composed of stars, would be completely re- 
solved by a further increase of optical power. In fact, this 
probability has almost been converted into a certainty by 



786 OUTLINES OF ASTRONOMY 

the magnificent reflecting telescope constructed by Lord 
Kosse, of six feet in aperture, which has resolved or ren- 
dered resolvable multitudes of nebula3 which had resisted 
all inferior powers. The sublimity of the spectacle afforded 
by that instrument of some of the larger globular and other 
clusters enumerated in the list given in art. 867, is declared 
by all who have witnessed it to be such as no words can 
express. 

(871.) Although, therefore, nebulae do exist, which even 
in this powerful telescope appear as nebulas, without any 
sign of resolution, it may very reasonably be doubted 
whether there be really any essential physical distinction 
between nebulas and clusters of stars, at least in the nature 
of the matter of which they consist, 8 and whether the dis- 
tinction between such nebulas as are easily resolved, barely 
resolvable with excellent telescopes, and altogether irre- 
solvable with the best, be anything else than one of degree, 
arising merely from the excessive minuteness and multitude 
of the stars, of which the latter, as compared with the 
former, consist. The first impression which Halley, and 
other early discoverers of nebulous objects received from 
their peculiar aspect, so different from the keen, concen- 
trated light of mere stars, was that of a phosphorescent 
vapor (like the matter of a comet's tail) or a gaseous and 
(so to speak) elementary form of luminous sidereal matter. 4 
Admitting the existence of such a medium, dispersed in 
some cases irregularly through vast regions in space, in 
others confined to narrower and more definite limits, Sir 
W. Herschel was led to speculate on its gradual subsidence 
and condensation by the effect of its own gravity, into 
more or less regular spherical or spheroidal forms, denser 



3 See Note K. 4 Halley, Pliil. Trans, xxix. p. 390. 



OUTLINES OF ASTRONOMY 787 

(as they must in that case be) toward the centre. Assuming 
that in the progress of this subsidence, local centres of con- 
densation, subordinate to the general tendency, would not 
be wanting, he conceived that in this way solid nuclei 
might arise, whose local gravitation still further condens- 
ing, and so absorbing the nebulous matter, each in its im- 
mediate neighborhood, might ultimately become stars, and 
the whole nebula finally take on the state of a cluster of 
stars. Among the multitude of nebulas revealed by his 
telescopes, every stage of this process might be considered 
as displayed to our eyes, and in every modification of form 
to which the general principle might be conceived to apply. 
The more or less advanced state of a nebula toward its 
segregation into discrete stars, and of these stars them- 
selves toward a denser state of aggregation round a central 
nucleus, would thus be in some sort an indication of age. 
Neither is there any variety of aspect which nebulas offer, 
which stands at all in contradiction to this view. Even 
though we should feel ourselves compelled to reject the idea 
of a gaseous or vaporous "nebulous matter," it loses little 
or none of its force. Subsidence, and the central aggrega- 
tion consequent on subsidence, may go on quite as well 
among a multitude of discrete bodies under the influence 
of mutual attraction, and feeble or partially opposing pro- 
jectile motions, as among the particles of a gaseous fluid. 

(872.) The "nebular hypothesis," as it has been termed, 
and the theory of sidereal aggregation stand, in fact, quite 
independent of each other, the one as a physical concep- 
tion of processes which may yet, for aught we know, have 
formed part of that mysterious chain of causes and effects 
antecedent to the existence of separate self-luminous solid 
bodies; the other, as an application of dynamical principles 



788 OUTLINES OF ASTRONOMY 

to cases of a very complicated nature no doubt, but in which, 
the possibility or impossibility, at least, of certain general 
results may be determined on perfectly legitimate princi- 
ples. Among a crowd of solid bodies of whatever size, 
animated by independent and partially opposing impulses, 
motions opposite to each other must produce collision, de- 
struction of velocity, and subsidence or near approach 
toward the centre of preponderant attraction; while those 
which conspire, or which remain outstanding after such 
conflicts, must ultimately give rise to circulation of a per- 
manent character. Whatever we may think of such colli- 
sions as events, there is nothing in this conception contrary 
to sound mechanical principles. It will be recollected that 
the appearance of central condensation among a multitude 
of separate bodies in motion, by no means implies per- 
manent proximity to the centre in each any more than 
the habitually crowded state of a market-place, to which 
a large proportion of the inhabitants of a town must fre- 
quently or occasionally resort, implies the permanent resi- 
dence of each individual within its area. It is a fact that 
clusters thus centrally crowded do exist, and therefore the 
conditions of their existence must be dynamically possible, 
and in what has been said we may at least perceive some 
glimpses of the manner in which they are so. The actual 
intervals between the stars, even in the most crowded parts 
of a resolved nebula, to be seen at all by us, must be enor- 
mous. Ages, which to us may well appear indefinite, may 
easily be conceived to pass without a single instance of 
collision, in the nature of a catastrophe. Such may have 
gradually become rarer as the system has emerged from 
what must be considered its chaotic state, till at length, in 
the fulness of time, and under the prearranging guidance 



OUTLINES OF ASTRONOMY 789 

of that Design which pervades universal nature, each indi- 
vidual may have taken up such a course as to annul the 
possibility of further destructive interference. 

(873.) But to return from the regions of speculation to 
the description of facts. Next in regularity of form to the 
globular clusters, whose consideration has led us into this 
digression, are elliptic nebulae, more or less elongated. 
And of these it may be generally remarked, as a fact un- 
doubtedly connected in some very intimate manner with 
the dynamical conditions of their subsidence, that such 
nebulae are, for the most part, beyond comparison more 
difficult of resolution than those of globular form. They 
are of all degrees of excentricity, from moderately oval 
forms to ellipses so elongated as to be almost linear, which 
are, no doubt, edge-views of very flat ellipsoids. In all of 
them the density increases toward the centre, and as a gen- 
eral law it may be remarked that, so far as we can judge 
from their telescopic appearance, their internal strata ap- 
proach more nearly to the spherical form than their exter- 
nal. Their resolvability, too, is greater in the central parts, 
whether owing to a real superiority of size in the central 
stars or to the greater frequency of cases of close juxta- 
position of individuals, so that two or three united appear 
as one. In some the condensation is slight and gradual, in 
others great and sudden; so sudden, indeed, as to offer the 
appearance of a dull and blotted star, standing in the midst 
of a faint, nearly equable elliptic nebulosity, of which two 
remarkable specimens occur in E. A. 12 h 10 m 33 s , N. P. D. 
41° 46', and in 13 h 27 m 28 3 , 119° 0' (1830). 

(874.) The largest and finest specimens of elliptic nebulae 
which the heavens afford are that in the girdle of Androm- 
eda (near the star v of that constellation) and that discov- 



T90 OUTLINES OF ASTRONOMY 

ered in 1783, by Miss Carolina Herschel, in R. A. h 39 m 12 8 , 
N. P. D. 116° 13'. The nebula in Andromeda (Plate II. 
fig. 3) is visible to the naked eye, and is continually mis- 
taken for a comet by those unacquainted with the heavens. 
Simon Marius, who noticed it in 1612 (though it appears also 
to have been seen and described as oval, in 995), describes 
its appearance as that of a candle shining through horn, and 
the resemblance is not inapt. Its form, as seen through or- 
dinary telescopes, is a pretty long oval, increasing by insen- 
sible gradations of brightness, at first very gradually, but at 
last more rapidly, up to a central point, which, though very 
much brighter than the rest, is decidedly not a star, but 
nebula of the same general character with the rest in a state 
of extreme condensation. Casual stars are scattered over it, 
but with a reflector of 18 inches in diameter, there is noth- 
ing to excite any suspicion of its consisting of stars. Ex- 
amined with instruments of superior defining power, how- 
ever, the evidence of its resolvability into stars may be 
regarded as decisive. Mr. G. P. Bond, assistant at the ob- 
servatory of Cambridge, U. S., describes and figures it as 
extending nearly 2\° in length, and upward of a degree in 
breadth (so as to include two other smaller adjacent nebulae), 
of a form, generally speaking, oval, but with a considerably 
protuberant irregularity at its north following extremity, 
very suddenly condensed at the nucleus almost to the sem- 
blance of a star, and though not itself clearly resolved, yet 
thickly sown over with visible minute stars, so numerous as 
to allow of 200 being counted within a field of 20' diameter 
in the richest parts. But the most remarkable feature in his 
description is that of two perfectly straight, narrow, and 
comparatively or totally obscure streaks which run nearly 
the whole length of one side of the nebula, and (though 



OUTLINES OF ASTRONOMY 791 

slightly divergent from each other) nearly parallel to its 
longer axis. These streaks (which obviously indicate a 
stratified structure in the nebula, if, indeed, they do not 
originate in the interposition of imperfectly transparent mat- 
ter between us and it) are not seen on a general and cursory 
view of the nebula; they require attention to distinguish 
them, 5 and this circumstance must be borne in mind when 
inspecting the very extraordinary engraving which illus- 
trates Mr. Bond's account. The figure given in our Plate 
11. Jig. 3, is from a rather hasty sketch, and makes no pre- 
tensions to exactness. A similar, but much more strongly 
marked case of parallel arrangement than that noticed by 
Mr. Bond in this, is one in which the two semiovals of an 
elliptically formed nebula appear cut asunder and separated 
by a broad obscure band parallel to the longer axis of the 
nebula, in the midst of which a faint streak of light parallel 
to the sides of the cut appears: it is seen in the southern 
hemisphere in R. A. 13 h 15 m 31 s , N. P. D. 132° 8' (1830). 
The nebulae in 12 h 27 m 3 s , 63° 5', and 12 h 31 m 11 s , 100° 40' 
present analogous features. 

(875.) Annular nebulae also exist, but are among the 
rarest objects in the heavens. The most conspicuous of 
this class is to be found almost exactly half way between 
(3 and y Lyrae, and may be seen with a telescope of moderate 
power. It is small and particularly well defined, so as to 
have more the appearance of a flat oval solid ring than 
of a nebula. The axes of the ellipse are to each other in 
the proportion of about 4 to 5, and the opening occupies 
about half or rather more than half the diameter. The 
central vacuity is not quite dark, but is filled in with faint 
nebula, like a gauze stretched over a hoop. The powerful 

5 Trans. American Acad., vol. iii. p. 80. 



792 



OUTLINES OF ASTRONOMY 



telescopes of Lord Eosse resolve this object into excessively 
minute stars, and show filaments of stars adhering to its 
edges. 6 

(876.) Planetary nebula are very extraordinary ob- 
jects. They have, as their name imports, a near, in some 
instances, a perfect resemblance to planets, presenting disks 
round, or slightly oval, in some quite sharply terminated, 
in others a little hazy or softened at the borders. Their 
light is in some perfectly equable, in others mottled and of 
a very peculiar texture, as if curdled. They are compara- 
tively rare objects, not above four or five and twenty hav- 
ing been hitherto observed, and of these nearly three- 
fourths are situated in the southern hemisphere. Being 
very interesting objects we subjoin a list of the most re- 
markable. 7 Among these may be more particularly speci- 
fied the sixth in order, situated in the Cross. Its light is 
about equal to that of a star of the 6*7 magnitude, its diam- 
eter about 12", its disk circular or very slightly elliptic, 
and with a clear, sharp, well-defined outline, having exactly 
the appearance of a planet with the exception only of its 





6 The places of 


some 


remarkable annular 


nebulae 


(for 


1830) 


are, 








R. 


A. 






N. P 


D. 




R. 


A. 






N. P 


. D. 


I. 


10* 


16m 


36 8 




107° 


48' 


4. 


17h 


19 m 


2s 




113° 


37' 


2. 


12 


42 


52 




47 


57 


5. 


18 


47 


13 




57 


11 


4. 


17 


10 


39 




128 


18 


6. 


20 


9 


33 




59 


57 



1 Places for 1830 of twelve of the most remarkable planetary nebulas. 


R. A. 


N. P. D. 


R. A. 


N. P. P. 


R. A. N. P. D. 


h. m. s. 

1. 7 34 2 

2. 9 16 39 

3. 9 59 52 

4. 10 16 36 


o 1 

104 20 
147 35 
129 36 
107 47 


h. m. s. 

5. 11 4 49 

6. 11 41 56 

7. 15 5 18 

8. 19 10 9 


o / 

34 4 
146 14 
135 1 

83 46 


h. m. s. ° ' j" 
9. 19 34 21 104 33 S 

10. 19 40 19 39 54 | 

11. 20 54 53 102 2 

12. 23 17 44 48 24 J 



OUTLINES OF ASTRONOMY 793 

color, which is a fine and full blue verging somewhat upon 
green. And it is not a little remarkable that this phenom- 
enon of a blue color, which is so rare among stars (except 
when in the immediate proximity of yellow stars) occurs, 
though less strikingly, in three other objects of this class, 
viz. in No. 4, whose color is sky-blue, and in Nos. 11 and 
12, where the tint, though paler, is still evident. Nos. 2, 
7, 9, and 12, are also exceedingly characteristic objects 
of this class. Nos. 3, 4, and 11 (the latter in the parallel 
of v Aquarii, and about 5 m preceding that star), are con- 
siderably elliptic, and (respectively) about 38", 30" and 15" 
in diameter. On the disk of No. 3, and very nearly in the 
centre of the ellipse, is a star 9m, and the texture of its 
light, being velvety, or as if formed of fine dust, clearly 
indicates its resolvability into stars. The largest of these 
objects is No. 5, situated somewhat south of the parallel 
of /? Ursse Majoris and about 12 m following that star. Its 
apparent diameter is 2' 40", which, supposing it placed at 
a distance from us not more than that of 61 Cygni, would 
imply a linear one seven times greater than that of the orbit 
of Neptune. The light of this stupendous globe is perfectly 
equable (except just at the edge, where it is slightly soft- 
ened), and of considerable brightness. Such an appearance 
would not be presented by a globular space uniformly filled 
with stars or luminous matter, which structure would nec- 
essarily give rise to an apparent increase of brightness 
toward the centre in proportion to the thickness traversed 
by the visual ray. We might, therefore, be induced to con- 
clude its real constitution to be either that of a hollow 
spherical shell or of a flat disk, presented to us (by a highly 
improbable coincidence) in a plane precisely perpendicular 
to the visual ray. 



794 OUTLINES OF ASTRONOMY 

(877.) Whatever idea we may form of the real nature of 
such a body, or of the planetary nebulae in general, which 
all agree in the absence of central condensation, it is evident 
that the intrinsic splendor of their surfaces, if continuous, 
must be almost infinitely less than that of the sun. A cir- 
cular portion of the sun's disk, subtending an angle of 1', 
would give a light equal to that of 780 full moons; while 
among all the objects in question there is not one which can 
be seen with the naked eye. M. Arago has surmised that 
they may possibly be envelopes shining by reflected light, 
from a solar body placed in their centre, invisible to us by 
the effect of its excessive distance ; removing, or attempting 
to remove the apparent paradox of such an explanation, by 
the optical principle that an illuminated surface is equally 
bright at all distances, and, therefore, if large enough to 
subtend a measurable angle, can be equally well seen, 
whereas the central body, subtending no such angle, has 
its effect on our sight diminished in the inverse' ratio of 
the square of its distance. 8 The immense optical powers 
applied by Lord Eosse and Mr. Lassell to the examination 
of these enigmatical objects have hitherto only added to 
the mystery which hangs about them, by disclosing caprices 
of structure in several of them of the most extraordinary 
nature. 9 

(878.) Double nebulas occasionally occur — and when such 

8 With due deference to so high an authority we must demur to the conclu- 
sion. Even supposing the envelope to reflect and scatter (equally in all direc- 
tions) half the light of the central sun, the portion of the light so scattered 
which would fall to our share could not exceed the remaining half which that 
sun itself would still send to us by direct radiation. But this, ex hypothesi, is 
too small to affect the eye with any luminous perception, when concentrated 
in a point, much less then could it do so if spread over a surface many million 
times exceeding in angular area the apparent disk of the central sun itself. 
(See Annuaire du Bureau des Longitudes, 1842, p. 409, 410, 411.) 

9 See the figures in their papers, Phil. Trans. 1850 and 1861, and Mem. Ast. 
Soc. vol. xxxvi. 



OUTLINES OF ASTRONOMY 795 

is the case, the constituents most commonly belong to the 
class of spherical nebulas, and are in some instances un- 
doubtedly globular clusters. All the varieties of double 
stars, in fact, as to distance, position, and relative bright- 
ness, have their counterparts in double nebulas; besides 
which the varieties of form and gradation of light in the 
latter afford room for combinations peculiar to this class 
of objects. Though the conclusive evidence of observed 
relative motion be yet wantiDg, and though from the vast 
scale on which such systems are constructed, and the prob- 
able extreme slowness of the angular motion, it may con- 
tinue for ages to be so, yet it is impossible, when we cast 
our eyes upon such objects, or on the figures which have 
been given of them, 10 to doubt their physical connection. 
The argument drawn from the comparative rarity of the 
objects in proportion to the whole extent of the heavens, 
so cogent in the case of the double stars, is infinitely more 
so in that of the double nebulas. Nothing more magnificent 
can be presented to our consideration, than such combina- 
tions. Their stupendous scale, the multitude of individuals 
.they involve, the perfect symmetry and regularity which 
many of them present, the utter disregard of complication 
in thus heaping together system upon system, and con- 
struction upon construction, leave us lost in wonder and 
admiration at the evidence they afford of infinite power 
and unfathomable design. 

(879.) Nebulas of regular forms often stand in marked 
and symmetrical relation to stars, both single and double. 
Thus we are occasionally presented with the beautiful and 
striking phenomenon of a sharp and brilliant star concen- 
trically surrounded by a perfectly circular disk or atmos= 

M Phil. Trans. 1833, Plate vii. 

Astronomy — Yol. XX — 15 



796 OUTLINES OF ASTRONOMY 

phere of faint light, in some cases dying away insensibly 
on all sides, in others almost suddenly terminated. These 
are Nebulous Stars. Fine examples of this kind are the 
45th and 69th nebulae of Sir fm, Herschel's fourth class 11 
(E. A. 7 h 19 m 8 s . N. P. D. 68° 45', and 3 h 58 m 36 s , 59° 40') in 
which stars of the 8th magnitude are surrounded by photo- 
spheres of the kind above described respectively of 12" and 
25" in diameter. Among stars of larger magnitudes, 55 
Andromedse and 8 Canum Yenaticorum may be named 
as exhibiting the same phenomenon with more brilliancy, 
but perhaps with less perfect regularity. 

(880.) The connection of nebulas with double stars is in 
many instances extremely remarkable. Thus in E. A. 18 h 
7 m I s , N. P. D. 109° 56', occurs an elliptic nebula having 
its longer axis about 50", in which, symmetrically placed, 
and rather nearer the vertices than the foci of the ellipse, 
are the equal individuals of a double star, each of the 10th 
magnitude. In a similar combination noticed by M. Struve 
(in E. A. 18 h 25 m , N. P. D. 25° 7'), the stars are unequal and 
situated precisely at the two extremities of the major axis. 
In E. A. 13 h 47 m 33 s , K P. D. 129° 9', an oval nebula of 2' 
in diameter has very near its centre a close double star, 
the individuals of which, slightly unequal, and about the 
9*10 magnitude, are not more than 2" asunder. The nucleus 
of Messier's 64th nebula is "strongly suspected" to be a 
close double star — and several other instances might be cited. 



11 The classes here referred to are not the species described in art. 868, but 
lists of nebulae, eight in number, arranged according to brightness, size, density 
of clustering, etc. , in one or other of which all nebulas were originally classed 
by him. Class I. contains "Bright nebulas"; II. "Faint do."; III. "Very 
faint do."; IV. "Planetary nebulas, stars with bars, milky chevelures, short 
rays, remarkable shapes, etc."; V. "Very large nebulas"; VI. "Very com- 
pressed rich clusters"; VII. "Pretty much compressed do. " ; VIII. "Coarsely 
scattered clusters." 



OUTLINES OF ASTRONOMY 797 

(881.) Among the nebulae which, though deviating more 
from symmetry of form, are yet not wanting in a certain 
regularity of figure, and which seem clearly entitled to be 
regarded as systems of a definite nature, however mysteri- 
ous their structure and destination, by far the most remark- 
able are the 27th and 51st of Messier's Catalogue. 13 The 
former consists of two round or somewhat oval nebulous 
masses united by a short neck of nearly the same density. 
Both this and the masses graduate off, however, into a 
fainter nebulous envelope which completes the figure into 
an elliptic form, of which the interior masses with their 
connection occupy the lesser axis. Seen in a reflector of 
18 inches in aperture, the form has considerable regularity; 
and though a few stars are here and there scattered over it, 
it is unresolved. Lord Rosse, viewing it with a reflector of 
double that aperture, describes and figures it as resolved 
into numerous stars with much intermixed nebula; while 
the symmetry of form, by rendering visible features too 
faint to be seen with inferior power, is rendered consider- 
ably less striking, though by no means obliterated. 

(882.) The 51st nebula of Messier, viewed through an 
18-inch reflector, presents the appearance of a large and 
bright globular nebula, surrounded by a ring at a consider- 
able distance from the globe, very unequal in brightness 
in its different parts, and subdivided through about two- 
fifths of its circumference as if into two laminae, one of 
which appears as if turned up toward the eye out of the 
plane of the rest. Near it (at about a radius of the ring 
distant) is a small bright round nebula. Viewed through 



12 Places for 1830: R. A. 19h. 52m. 12s., N. P. D. 67° 44', and R. A. 13h. 
22m. 59s., N. P. D. 41° 56'. 



798 OUTLINES OF ASTRONOMY 

the 6-feet reflector of Lord Eosse the aspect is much altered. 
The interior, or what appeared the upturned portion of the 
ring, assumes the aspect of a nebulous coil or convolution 
tending in a spiral form toward the centre, and a general 
tendency to a spiroid arrangement of the streaks of nebula 
connecting the ring and central mass which this power 
brings into view, becomes apparent, and forms a very 
striking feature. The outlying nebula is connected by 
a narrow nebulous arc with the ring, and the whole has a 
resolvable character. (See Plate VI. fig. 3.) Both Lord 
Eosse and Mr. Lassell have found this spiral character, 
even still more marked, to belong to many other nebulas: 
sufficiently numerous, in fact, to form a class apart, of 
which Messier's 99th nebula is a fine specimen. 

(883.) "We come now to a class of nebulas of totally 
different character. They are of very great extent, utterly 
devoid of all symmetry of form — on the contrary, irregular 
and capricious in their shapes and convolutions to a most 
extraordinary degree, and no less so in the distribution of 
their light. No two of them can be said to present any 
similarity of figure or aspect, but they have one important 
character in common. They are all situated in, or very 
near, the borders of the Milky Way. The most remote 
from it is that in the sword-handle of Orion, which being 
20° from the galactic circle, and 15° from the visible border 
of the Via Lactea, might seem to form an exception, though 
not a striking one. But this very situation may be adduced 
as a corroboration of the general view which this principle 
of localization suggests. For the place in question is situ- 
ated in the prolongation of that faint offset of the Milky 
Way which we traced (art. 787) from a and e Persei toward 
Aldebaran and the Hyades, and in the zone of Great Stars 



OUTLINES OF ASTRONOMY 799 

noticed in art. 785 as an appendage of, and probably bear- 
ing relation to that stratum. 

(88i.) From this it would appear to follow, almost as a 
matter of course, that they must be regarded as outlying, 
very distant, and as it were detached fragments of the great 
stratum of the Galaxy, and this view of the subject is 
strengthened when we find on mapping down their places 
that they may all be grouped in four great masses or nebu- 
lous regions — that of Orion, of Argo, of Sagittarius, and of 
Cygnus. And thus, inductively, we may gather some in- 
formation respecting the structure and form of the Galaxy 
itself, which, could we view it as a whole, from a distance 
such as that which separates us from these objects, would 
very probably present itself under an aspect quite as com- 
plicated and irregular. 

(885.) The great nebula surrounding the stars marked 
6 1 in the sword-handle of Orion was discovered by Huy- 
ghens in 1656, and has been repeatedly figured and described 
by astronomers since that time. Its appearance varies 
greatly (as that of all nebulous objects does) with the in- 
strumental power applied, so that it is difficult to recognize 
in representations made with inferior telescopes, even prin- 
cipal features, to say nothing of subordinate details. Until 
this became well understood, it was supposed to have 
changed very materially, both in form and extent, during 
the interval elapsed since its first discovery. No doubt, 
however, now remains that these supposed changes have 
originated partly from the cause above mentioned, partly 
from the difficulty of correctly drawing, and, still more, 
engraving such objects, and partly from a want of sufficient 
care in the earlier delineators themselves in faithfully copy- 
ing that which they really did see. Our figure (Plate IV, 



800 OUTLINES OF ASTRONOMY 

Jig. 1) is reduced from a larger one made under very favor- 
able circumstances, from drawings taken with an 18-inch 
reflector at the Cape of Good Hope, where its meridian 
altitude greatly exceeds what it has at European stations. 
The area occupied by this figure is about one 25th part of 
a square degree, extending in E. A. (or horizontally) 2 m 
of time, equivalent almost exactly to 30' in arc, the object 
being very near the equator, and 24' vertically, or in polar 
distance. The figure shows it reversed in declination, the 
northern side being lowermost, and the preceding toward 
the right hand. In form, the brightest portion offers a re- 
semblance to the head and yawning jaws of some monstrous 
animal, with a sort of proboscis running out from the snout. 
Many stars are scattered over it, which for the most part 
appear to have no connection with it, and the remarkable 
sextuple star d 1 Orionis, of which mention has already been 
made (art. 837), occupies a most conspicuous situation close 
to the brightest portion, at almost the edge of the opening 
of the jaws. It is remarkable, however, that within the 
area of the trapezium no nebula exists. The general aspect 
of the less luminous and cirrous portion is simply nebulous 
and irresolvable, but the brighter portion immediately ad- 
jacent to the trapezium, forming the square front of the 
head, is shown with the 18-inch reflector broken up into 
masses (very imperfectly represented in the figure), whose 
mottled and curdling light evidently indicates by a sort of 
granular texture its consisting of stars, and when examined 
under the great light of Lord Eosse's reflector, or the ex- 
quisite defining power of the great achromatic at Cambridge, 
U. S., is evidently perceived to consist of clustering stars. 
There can therefore be little doubt as to the whole consist- 
ing of stars, too minute to be discerned individually even 



OUTLINES OF ASTRONOMY 801 

with these powerful aids, but which become visible as 
points of light when closely adjacent in the more crowded 
parts in the mode already more than once suggested. 

(886.) The nebula is not confined to the limits of our 
figure. Northward of about 33', and nearly on the same 
meridian are two stars marked C 1 and C 2 Ononis, in- 
volved in a bright and branching nebula of very singular 
form, and south of it is the star c Orionis, which is also in- 
volved in strong nebula. Careful examination with power- 
ful telescopes has traced out a continuity of nebulous light 
between the great nebula and both these objects, and there 
can be little doubt that the nebulous region extends north- 
ward, as far as e in the belt of Orion, which is involved in 
strong nebulosity, as well as several smaller stars in the 
immediate neighborhood. Professor Bond has given a 
beautiful figure of the great nebula in Trans. American 
Acad, of Arts and Sciences, new series, vol. iii., and Lord 
Oxmantown a superb one in Phil. Tr. 1868. 

(887.) The remarkable variation in lustre of the bright 
star f) in Argo, has been already mentioned. This star is 
situated in the most condensed region of a very extensive 
nebula or congeries of nebular masses, streaks and branches, 
a portion of which is represented in fig. 2, Plate IY. The 
whole nebula is spread over an area of fully a square degree 
in extent, of which that included in the figure occupies 
about one-fourth, that is to say, 28' in polar distance, and 
32' of arc in E. A., the portion not included being, though 
fainter, even more capriciously contorted than that here 
depicted, in which it should be observed that the preceding 
side is toward the right hand, and the southern uppermost. 
Viewed with an 18-inch reflector, no part of this strange 
object shows any sign of resolution into stars, nor in the 



802 OUTLINES OF ASTRONOMY 

brightest and most condensed portion adjacent to the sin- 
gular oval vacancy in the middle of the figure is there any 
of that curdled appearance or that tendency to break up 
into bright knots with intervening darker portions which 
characterize the nebula of Orion, and indicate its resolva- 
bility. The whole is situated in a very rich and brilliant 
part of the Milky Way, so thickly strewed with stars 
(omitted in the figure), that in the rrea occupied by the 
nebula, not less than 1200 have been actually counted, 
and their places in B,. A. and P. D. determined. Yet it is 
obvious that these have no connection whatever with the 
nebula, being, in fact, only a simple continuation over it of 
the general ground of the galaxy, which on an average 
of two hours in right ascension in this period of its course 
contains no less than 3138 stars to the square degree, all, 
however, distinct, and (except where the object in question 
is situated) seen projected on a perfectly dark heaven, with- 
out any appearance of intermixed nebulosit} 7 . The conclu- 
sion can hardly be avoided, that in looking at it we see 
through, and beyond the Milky Way, far out into space, 
through a starless region, disconnecting it altogether from 
our system. "It is not easy for language to convey a full 
impression of the beauty and sublimity of the spectacle 
which this nebula offers, as it enters the field of view of 
a telescope fixed in right ascension, by the diurnal mo- 
tion, ushered in as it is by so glorious and innumerable 
a procession of stars, to which it forms a sort of climax," 
and in a part of the heavens otherwise full of interest. One 
other bright and very remarkably formed nebula of con- 
siderable magnitude precedes it nearly on the same paral- 
lel, but without any traceable connection between them. 
(888.) !The nebulous group of Sagittarius consists of sev- 



OUTLINES OF ASTRONOMY 803 

eral conspicuous nebuhe 13 of very extraordinary forms by 
no means easy to give an idea of by mere description. One 
of them (A, 1991 14 ) is singularly trifid, consisting of three 
bright and irregularly formed nebulous masses, graduating 
away insensibly externally, but coming up to a great inten- 
sity of light at their interior edges, where they inclose and 
surround a sort of three-forked rift, or vacant area, abruptly 
and uncouthly crooked, and quite void of nebulous light. 
A beautiful triple star is situated precisely on the edge of 
one of these nebulous masses just where the interior vacancy 
forks out into two channels. A fourth nebulous mass 
spreads like a fan or downy plume from a star at a little 
distance from the triple nebula. 

(889.) Nearly adjacent to the last described nebula, and 
no doubt connected with it, though, the connection has not 
yet been traced, is situated the 8th nebula of Messier's 
Catalogue. It is a collection of nebulous folds and masses, 
surrounding and including a number of oval dark vacancies, 
and in one place coming up to so great a degree of bright- 
ness, as to offer the appearance of an elongated nucleus. 
Superposed upon this nebula, and extending in one direc- 
tion beyond its area, is a fine and rich cluster of scattered 
stars, which seem to have no connection with it, as the 
nebula does not, as in the region of Orion, show any ten- 
dency to congregate about the stars. 

(890.) The 19th nebula of Messier's Catalogue, though 

13 About R. A. 17h. 52m., N. P. D. 113° 1', four nebulae, No. 41 of Sir 
"Win. Herschel's 4th class, and Nos. 1, 2, 3, of his 5th all connected into one 
great complex nebula.— In R, A. 17h. 53m. 2ts., N. P. D. 114° 21', the 8th, 
and in 18h. 11m., 106° 15', the 17 th of Messier's Catalogue. 

14 This number refers to the catalogue of nebulae in Phil. Trans. 1833. 
The reader will find figures of the several nebulas of this group in that volume, 
plate iv. fig. 35, in the Author's "Results of Observations, etc., at the Cape 
of Good Hope." Plates i. fig. 1, and ii. figs. 1 and 2, and in Mason's Memoir 
in the collection of the American Phil. Soc, vol. vii. art. xiii. 



804 OUTLINES OF ASTRONOMY 

some degrees remote from the others, evidently belongs to 
this group. Its form is very remarkable, consisting of two 
loops like capital Greek Omegas, the one bright, the other 
exceedingly faint, connected at their bases by a broad and 
very bright band of nebula, insulated within which by 
a narrow comparatively obscure border, stands a bright, 
resolvable knot, or what is probably a cluster of exceed- 
ingly minute stars. A very faint round nebula stands in 
connection with the upper or convex portion of the brighter 
loop. 

(891.) The nebulous group of Cygnus consists of several 
large and irregular nebula?, one of which passes through the 
double star h Cygni, as a long, crooked, narrow streak, 
forking out in two or three places. The others, 15 observed 
in the first instance by Sir W. Herschel and by the author 
of this work as separate nebulae, have been traced into con- 
nection by Mr. Mason, and shown to form part of a curious 
and intricate nebulous system, consisting, 1st, of a long, 
narrow, curved, and forked streak, and, 2dly, of a cellular 
effusion of great extent, in which the nebula occurs inter- 
mixed with, and adhering to stars around the borders of the 
cells, while their interior is free from nebula, and almost 
so from stars. 

(892.) The Magellanic clouds, or the nubecula (major 
and minor), as they are called in the celestial maps and 
charts, are, as their name imports, two nebulous or cloudy 
masses of light, conspicuously visible to the naked eye, in 
the southern hemisphere, in the appearance and brightness 
of their light not unlike portions of the Milky Way of the 
same apparent size. They are, generally speaking, round, 
or somewhat oval, and the larger, which deviates most 

16 R. A. 20h. 49m. 20s„ N. P. D. 58° 2V. 



OUTLINES OF ASTRONOMY 805 

from the circular form, exhibits the appearance of an axis 
of light, very ill defined, and by no means strongly distin- 
guished from the general mass, which seems to open out 
at its extremities into somewhat oval sweeps, constituting 
the preceding and following portions of its circumference. 
A small patch, visibly brighter than the general light 
around, in its following part, indicates to the naked eye 
the situation of a very remarkable nebula (No. 30 Doradus 
of Bode's Catalogue), of which more hereafter. The greater 
nubecula is situated between the meridians of 4 h 40 m and 
6 h m and the parallels of 156° and 162° of N. P. D., and 
occupies an area of about 42 square degrees. The lesser, 
between the meridians 16 28 and l h 15 m and the parallels 
of 162° and 165° 1ST. P. D., covers about ten square degrees. 
Their degree of brightness may be judged of from the effect 
of strong moonlight, which totally obliterates the lesser, but 
not quite the greater. 

(893.) When examined through powerful telescopes, the 
constitution of the nubeculae, and especially of the nubecula 
major, is found to be of astonishing complexity. The gen- 
eral ground of both consists of large tracts and patches of 
nebulosity in every stage of resolution, from light, irresolv- 
able with 18 inches of reflecting aperture, up to perfectly 
separated stars like the Milky Way, and clustering groups 
sufficiently insulated and condensed to come under the 
designation of irregular, and in some cases pretty rich 
clusters. But besides those, there are also nebulas in abun- 
dance, both regular and irregular; globular clusters in every 
state of condensation; and objects of a nebulous character 
quite peculiar, and which have no analogue in any other 
region of the heavens. Such is the concentration of these 

16 It is laid down nearly an hour wrong in all the celestial charts and globes. 



806 OUTLINES OF ASTRONOMY 

objects, that in the area occupied by the nubecula major, 
not fewer than 278 nebulas and clusters have been enumer- 
ated, besides 50 or 60 outliers, which (considering the gen- 
eral barrenness in such objects of the immediate neighbor- 
hood) ought certainly to be reckoned as its appendages, 
being about 6\ per square degree, which very far exceeds 
the average of any other, even the most crowded parts of 
the nebulous heavens. In the nubecula minor, the concen- 
tration of such objects is less, though still very striking, 
37 having been observed within its area, and 6 adjacent, 
but outlying. The nubeculse, then, combine, each within 
its own area, characters which in the rest of the heavens 
are no less strikingly separated — viz. those of the galactic 
and the nebular system. Grlobular clusters (except in one 
region of small extent) and nebulae of regular elliptic forms 
are comparatively rare in the Milky Way, and are found 
congregated in the greatest abundance in a part of the 
heavens, the most remote possible from that circle; whereas, 
in the nubeculae, they are indiscriminately mixed with the 
general starry ground, and with irregular though small 
nebulas. 

(894.) This combination of characters, rightly considered, 
is in a high degree instructive, affording an insight into the 
probable comparative distance of stars and nebulce, and the 
real brightness of individual stars as compared one with 
another. Taking the apparent semidiameter of the nubec- 
ula major at 3°, and regarding its solid form as, roughly 
speaking, spherical, its nearest and most remote parts 
differ in their distance from us by a little more than a 
tenth part of our distance from its centre. The brightness 
of objects situated in its nearer portions, therefore, cannot 
be much exaggerated, nor that of its remoter much enfee- 



OUTLINES OF ASTRONOMY 807 

bled, by their difference of distance; yet within this glob- 
ular space we have collected upward of 600 stars of the 
7th, 8th, 9th, and 10th magnitudes, nearly 300 nebulas, 
and globular and other clusters, of all degrees of resolubility, 
and smaller scattered stars innumerable of every inferior 
magnitude, from the 10th to such as by their multitude and 
minuteness constitute irresolvable nebulosity, extending over 
tracts of many square degrees. Were there but one such 
object, it might be maintained without utter improbability 
that its apparent sphericity is only an effect of foreshorten- 
ing, and that in reality a much greater proportional differ- 
ence of distance between its nearer and more remote parts 
exists. But such an adjustment, improbable enough in one 
case, must be rejected as too much so for fair argument in 
two. It must, therefore, be taken as a demonstrated fact, 
that stars of the 7th or 8th magnitude and irresolvable 
nebula may co-exist within limits of distance not differing 
in proportion more than as 9 to 10, a conclusion which must 
inspire some degree of caution in admitting, as certain, many 
of the consequences which have been rather strongly dwelt 
upon in the foregoing pages. 

(895.) Immediately preceding the centre of the nubecula 
minor, and undoubtedly belonging to the same group, occurs 
the superb globular cluster, No. 47 Toucani of Bode, very 
visible to the naked eye, and one of the finest objects of this 
kind in the heavens. It consists of a very condensed spheri- 
cal mass of stars, of a pale rose color, concentrically inclosed 
in a much less condensed globe of white ones, 15' or 20' in 
diameter. This is the first in order of the list of such clus- 
ters in art. 867. 

(896.) Within the nubecula major, as already mentioned, 
and faintly visible to the naked eye, is the singular nebula 



808 OUTLINES OF ASTRONOMY 

(marked as the star 30 Doradiis in Bode's Catalogue) noticed 
by Lacaille as resembling the nucleus of a small comet. 
It occupies about one-500th part of the whole area of the 
nubecula, and is so satisfactorily represented in Plate Y. 
fig. 1, as to render further description superfluous. See 
art. 896 a, Note K. 

(897.) We shall conclude this chapter by the mention 
of two phenomena, which seem to indicate the existence of 
some slight degree of nebulosity about the sun itself, and 
even to place it in the list of nebulous stars. The first is 
that called the zodiacal light, which may be seen any very 
clear evening soon after sunset, about the months of March, 
April, and May, or at the opposite seasons before sunrise, 
as a cone or lenticularly-shaped light, extending from the 
horizon obliquely upward, and following generally the course 
of the ecliptic, or rather that of the sun's equator. The ap- 
parent angular distance of its vertex from the sun varies, 
according to circumstances, from 40° to 90°, and the breadth 
of its base perpendicular to its axis from 8° to 30°. It is 
extremely faint and ill defined, at least in this climate, 
though better seen in tropical regions, but cannot be mis- 
taken for any atmospheric meteor or aurora borealis. It is 
manifestly in the nature of a lenticularly-formed envelope, 17 
surrounding the sun, and extending beyond the orbits of 
Mercury and Yenus, and nearly, perhaps quite, attaining 
that of the earth, since its vertex has been seen fully 90° 
from the sun's place in a great circle. It may be conjec- 
tured to be no other than the denser part of that medium, 



11 I cannot imagine upon what grounds Humboldt persists in ascribing to it 
the form of a ring encircling the sun. For a most elaborate series of observa- 
tions of the zodiacal light, by the Rev. G-. Jones, see "United States Japan 
Expedition," vol. iii. 4°. Washington, 1856. It contains 357 plates of its 
appearance. 



OUTLINES OF ASTRONOMY 809 

which, we have some reason to believe, resists the motion 
of comets; loaded, perhaps, with the actual materials of the 
tails of millions of those bodies, of which thej have been 
stripped in their successive perihelion passages (art. 566). 
An atmosphere of the snn, in any proper sense of the word, 
it cannot be, since the existence of a gaseous envelope 
propagating pressure from part to part; subject to mutual 
friction in its strata, and therefore rotating in the same or 
nearly the same time with the central body; and of such 
dimensions and ellipticity, is utterly incompatible with 
dynamical laws. If its particles have inertia, they must 
necessarily stand with respect to the sun in the relation of 
separate and independent minute planets, each having its 
own orbit, plane of motion, and periodic time. The total 
mass being almost nothing compared to that of the sun, 
mutual perturbation is out of the question, though collisions 
among such as may cross each other's paths may operate in 
the course of indefinite ages to effect a subsidence of at least 
some portion of it into the body of the sun or those of the 
planets. 

(898.) Nothing prevents that these particles, or some 
among them, may have some tangible size, and be at very 
great distances from each other. Compared with planets 
visible in our most powerful telescopes, rocks and stony 
masses of great size and weight would be but as the im- 
palpable dust which a sunbeam renders visible as a sheet 
of light when streaming through a narrow chink into a dark 
chamber. It is a fact, established by the most indisputable 
evidence, that stony masses and lumps of iron do occasion- 
ally, and indeed by no means infrequently, fall upon the 
earth from the higher regions of our atmosphere (where it 
is obviously impossible they can have been generated), and 



810 OUTLINES OF ASTRONOMY 

that they have done so from the earliest times of history. 
Four instances are recorded of persons being killed by their 
fall. A block of stone fell at ^Egos Potamos, B.C. 465, as 
large as two millstones; another at Narni, in 921, projected, 
like a rock, four feet above the surface of the river, into 
which it was seen to fall. The emperor Jehangire had a 
sword forged from a mass of meteoric iron which fell, in 
1620, at Jalandar, in the Punjab. 18 Sixteen instances of 
the fall of stones in the British Isles are well authenticated 
to have occurred since 1620, one of them in London. In 
1803, on the 26th of April, thousands of stones were scat- 
tered by the explosion into fragments of a large fiery globe 
over a region of twenty or thirty square miles around the 
town of L'Aigle, in Normandy. The fact occurred at mid- 
day, and the circumstances were officially verified by a com- 
mission of the French government. 19 These, and innumer- 
able other instances, 20 fully establish the general fact; and 
after vain attempts to account for it by volcanic projection, 
either from the earth or the moon, the planetary nature of 
these bodies seems at length to be almost generally ad- 
mitted. The heat which they possess when fallen, the 
igneous phenomena which accompany them, their explo- 
sion on arriving within the denser regions of our atmos- 
phere, etc., are all sufficiently accounted for on physical 
principles, by the condensation of the air before them in 
consequence of their enormous velocity, and by the rela- 
tions of air in a highly attenuated state to heat. 81 

18 See the emperor's own very remarkable account of the occurrence, trans- 
lated in Phil. Trans. 1793, p. 202. 

19 See M. Biot's report in Mem. de PInslitut. 1806. 

20 See a list of upward of 200, published by Chladni, Annales du Bureau des 
Longitudes de France, 1825. 

21 Edinburgh Review, Jan. 1848, p. 195. It is very remarkable that no new 
chemical element has been detected in any of the numerous meteorolites which 
have been subjected to analysis. 



OUTLINES OF ASTRONOMY 811 

(S99.) Besides stony and metallic masses, however, it is 
probable that bodies of very different natures, or at least 
states of aggregation, are thus circulating round the sun. 
Shooting stars, often followed by long trains of light, and 
those great fiery globes, of more rare, but not very un- 
common occurrence, which are seen traversing the upper 
regions of our atmosphere, sometimes leaving trains behind 
them remaining for many minutes, sometimes bursting with 
a loud explosion, sometimes becoming quietly extinct, may 
not unreasonably be presumed to be bodies extraneous to 
our planet, which only become visible on becoming ignited 
in the act of grazing the confines of our atmosphere. 
Among the last-mentioned meteors, however, are some 
which can hardly be supposed solid masses. The remark- 
able meteor of August 18, 1783, traversed the whole of 
Europe, from Shetland to Eome, with a velocity of about 
30 miles per second, at a height of 50 miles from the surface 
of the earth, with a light greatly surpassing that of the full 
moon, and a real diameter of fully half a mile. Yet with 
these vast dimensions, it made a sudden bend in its course; 
it changed its form visibly, and at length quietly separated 
into several distinct bodies, accompanying each other in 
parallel courses, and each followed by a tail or train. 

(900.) There are circumstances in the history of shooting 
stars, which very strongly corroborate the idea of their ex- 
traneous or cosmical origin, and their circulation round the 
sun in definite orbits. On several occasions they have been 
observed to appear in unusual, and, indeed, astonishing 
numbers, so as to convey the idea of a shower of rockets, 
and brilliantly illuminating the whole heavens for hours 
together, and that not in one locality, but over whole con- 
tinents and oceans, and even (in one instance) in both hemi- 



812 OUTLINES OF ASTRONOMY 

spheres. Now it is extremely remarkable that, whenever 
this great display has been exhibited (at least in modern 
times), it has uniformly happened on the night between the 
12th and 13th, or on that between the 13th and 14th of 
November. Such cases occurred in 1799, 1832, 1833, and 
1834. On tracing back the records of similar phenomena, 
it has been ascertained, moreover, that more often those 
identical nights, but sometimes those immediately adjacent, 
have been, time out of mind, habitually signalized by such 
exhibitions. Another annually recurring epoch, in which, 
though far less brilliant, the display of meteors is more 
certain (for that of November is often interrupted for a 
great many years), is that of the 10th of August, on which 
night, and on the 9th and 11th, numerous, large, and bright 
shooting stars, with trains, are almost sure to be seen. 
Other epochs of periodic recurrence, less marked than the 
above, have also been to a certain extent established. 

(901.) It is impossible to attribute such a recurrence of 
identical dates of very remarkable phenomena to accident. 
Annual periodicity, irrespective of geographical position, 
refers us at once to the place occupied by the earth in its 
annual orbit, and leads direct to the conclusion that at that 
place the earth incurs a liability to frequent encounters or 
concurrences with a stream of meteors in their progress of 
circulation round the sun. Let us test this idea by pur- 
suing it into some of its consequences. In the first place 
then, supposing the earth to plunge, in its yearly circuit, 
into a uniform ring of innumerable small meteor-planets, 
of such breadth as would be traversed by it in one or two 
days; since during this small time the motions, whether of 
the earth or of each individual meteor, may be taken as 
uniform and rectilinear, and those of all the latter (at the 



OUTLINES OF ASTRONOMY 813 

place and time) parallel, or very nearly so, it will follow- 
that the relative motion of the meteors referred to the earth 
as at rest, will be also uniform, rectilinear, and parallel. 
Viewed, therefore, from the centre of the earth (or from any 
point in its circumference, if we neglect the diurnal velocity 
as very small compared with the annual) they will all appear 
to diverge from a common point, fixed in relation to the celes- 
tial sphere, as if emanating from a sidereal apex (art. 115). 

(902.) Now this is precisely what actually happens. The 
meteors of the 12th-ll:th of November, or at least the vast 
majority of them, describe apparently arcs of great circles, 
passing through or near y Leonis. No matter what the 
situation of that star with respect to the horizon or to its 
east and west points may be at the time of observation, 
the paths of the meteors all appear to diverge from that 
star. On the 9th-llth of August the geometrical fact is the 
same, the apex only differing; B Camelopardali being for 
that epoch the point of divergence. As we need not sup- 
pose the meteoric ring coincident in its plane with the 
ecliptic, and as for a ring of meteors we may substitute 
an elliptic annulus of any reasonable excentricity, so that 
both the velocity and direction of each meteor may differ 
to any extent from the earth's, there is nothing in the great 
and obvious difference in latitude of these apices at all mili- 
tating against the conclusion. 

(903.) If the meteors be uniformly distributed in such a 
ring or elliptic annulus, the earth's encounter with them 
in every revolution will be certain, if it occur once. But 
if the ring be broken, if it be a succession of groups re- 
volving in an ellipse in a period not identical with that of 
the earth, years may pass without a rencounter; and when 
such happen, they may differ to any extent in their inten- 



814 OUTLINES OF ASTRONOMY 

sity of character, according as richer or poorer groups have 
been encountered. 

(904.) No other plausible explanation of these highly 
characteristic features (the annual periodicity and diver- 
gence from a common apex, always alike for each respective 
epoch) has been even attempted, and accordingly the opinion 
is generally gaining ground among astronomers that shoot- 
ing stars belong to their department of science, and great 
interest is excited in their observation and the further 
development of their laws. The first connected and syste- 
matic series of observations of them, having for their object 
to trace out their relative paths with respect to the earth, 
are those of Benzenberg and Brandes, who, by noting the 
instants and apparent places of appearance and extinction, 
as well as the precise apparent paths among the stars, of 
individual meteors, from the extremities of a measured base 
line nearly 50,000 feet in length, were led to conclude that 
their heights at the instant of their appearance and disap- 
pearance vary from 16 miles to 140, and their relative 
velocities from 18 to 36 miles per second, velocities so 
great as clearly to indicate an independent planetary circu- 
lation round the sun. [A very remarkable meteor or bolide, 
which appeared on the 19th August, 1847, was observed at 
Dieppe and at Paris, with sufficient precision to admit of 
calculation of the elements of its orbit in absolute space. 
This calculation has been performed by M. Petit, director 
of the observatory of Toulouse, and the following hyperbolic 
elements of its orbit round the sun are stated by him (Astr. 
Nachr. 701) as its result; viz. Semimajor axis = —0-3240083; 
excentricity = 3*95130; perihelion distance = 0*95626; incli- 
nation to plane of the earth's equator, 18° 20' 18"; ascending 
node on the same plane, 10° 34' 48"; motion direct. Ac- 



OUTLINES OF ASTRONOMY 815 

cording to this calculation, the body would have occupied 
no less than 37340 years in travelling from the distance of 
the nearest fixed star supposed to have a parallax of 1".] 

(905.) It is by no means inconceivable that the earth ap- 
proaching to such as differ but little from it in direction 
and velocity, may have attached many of them to it as 
permanent satellites, and of these there may be some so 
large, and of such texture and solidity, as to shine by re- 
flected light, and become visible (such, at least, as are very 
near the earth) for a brief moment, suffering extinction by 
plunging into the earth's shadow; in other words, under- 
going total eclipse. Sir John Lubbock is of opinion that 
such is the case, and has given geometrical formulae for 
calculating their distances from observations of this nature. 22 
The observations of M. Petit would lead us to believe in 
the existence of at least one such body, revolving round the 
earth, as a satellite, in about 3 hours 20 minutes, and there- 
fore at a distance equal to 2 -513 radii of the earth from its 
centre, or 5,000 miles above its surface. 23 (See Note N.) 

(905 a.) In art. 400 the generation of heat by friction is 
suggested as affording a possible explanation of the supply 
of solar heat, without actual combustion. A very old doc- 
trine, advocated on grounds anything rather than reasonable 
or even plausible by Bacon, but afterward worked into 
a circumstantial and elaborate theory by the elder Seguin, 
which makes heat to consist in a continual, rapid, vibratory 
or gyratory motion of the particles of bodies, has of late 
been put forward into great prominence by Messrs. Mayer 
and Joule and Sir W. Thomson; according to this theory 
motion once generated or however originating, is never 

22 Phil. Mag., Lond. Ed. Dub. 1848, p. 80. 

33 Comptes Reudus, Oct. 12, 1846, and Aug. 9, 184*7. 



816 OUTLINES OF ASTRONOMY 

destroyed, but continues to subsist in the form of "vis 
viva" among the molecules of bodies, even when by their 
impact or mutual obstruction they appear to have been 
brought to rest. 24 The "vis viva" only takes another form, 
and is disseminated, as increased vibratory or gyratory 
movement, among their molecules; as such it is heat, or 
light, or both, and is communicated to the molecules of the 
luminiferous ether, and so distributed throughout that 
ether, constituting the phenomena of radiant light and 
heat. Granting a few postulates (not very easy of concep- 
tion, and still less so of admission when conceived), this 
theory is not without its plausibility, and certainly does 
(on its own conventions) give a consistent account of the 
production of heat by friction and impact. It has been ap- 
plied by Mr. Waterson and Sir W. Thomson to explain the 
evolution of solar light and heat, as follows. According to 
the former, the meteorolites which, revolving in very excen- 
tric or cometic orbits, arrive within the limits of the solar 
atmosphere are precipitated on the sun's surface in such 
abundance, and with such velocity, as to generate in the 
way above described the totality of the emitted radiants. 
Sir W. Thomson, undismayed as would appear by the per- 
petual battery thus kept up on the sun's surface (on every 
square foot of which, on Mr. Waterson's view of the sub- 
ject, a weight of matter equal to 5 lbs. would require to be 
delivered per hour with a velocity of 390 miles per second, 
covering the whole surface with stony or other solid ma- 
terial, to the depth of 12 feet per annum, if of the density 
of granite), prefers to consider the nebula of the zodiacal 
light in a vaporous state as continually subsiding into the 

24 On this point see a paper by the Author on the absorption of light, Lond. 
and Ed. Phil. Mag. -and Journ., 3d series, vol. iii. No. 18, Dec. 1833. 



OUTLINES OF ASTRONOMY 817 

sun, by gradual spiral approach, until suddenly meeting 
with greatly increased resistance in its atmosphere (as ar- 
riving in a state of more rapid revolution) by friction on the 
external envelope or photosphere of its surface (art. 389), pro- 
duces there the heat and light actually observed; whereas 
the theory of Mr. Waterson would place its origin on the 
solid surface itself, contrary to the observed fact. 96 Our 
readers will judge for themselves what degree of support 
the telescopic aspect of the sun's surface as described in 
arts. 386-395, and especially 387 a, Note Gr, affords. 



25 The quantity of matter annually required to be deposited on the sun, 
whether in a pulverulent, liquid, or vaporous form, by Sir W. Thomson's the- 
ory, is nearly double of that called for by Mr. Waterson's, viz. 24 feet of granite 
per annum, i.e. a mile in 260 years; so that the sun's apparent diameter would 
be increasing at the rate of about 1" per 100,000 years on this hypothesis. 

In the '"Manuel de la Science, ou Annuaire du Cosmos" for 1859, by the 
Abbe Moigno (a work of high interest, and, generally speaking, of great impar- 
tiality in the discussion of claims to scientific priority), pp. 85, 6, 7, 2 me partie, 
this article is so translated (probably for want of a perfect appreciation of the 
force of the expressions used in it) as to convey an unqualified adhesion to 
the theory in question and to M. Seguin's doctrine. This, however (especially 
the latter, as stated at length in Pt. I. pp. 224 et seq.), I am very far from 
prepared to give: — and the English reader will, I presume, consider the terms 
employed quite sufficiently guarded, even as respects the general principle; to 
Bay nothing of the specialties of M. Seguin's theory. — [Note added, 1859.] 



818 OUTLINES OF ASTRONOMY 



PART IV 

OF TEE ACCOUNT OF TIMS 

CHAPTER XVIII 

Natural Units of Time— Relation of the Sidereal to the Solar Day Affected 
by Precession — Incommensurability of the Day and Year— Its Incon- 
venience — How Obviated — The Julian Calendar — Irregularities at its 
first Introduction — Reformed by Augustus — Gregorian Reformation — 
Solar and Lunar Cycles — Indiction — Julian Period — Table of Chrono- 
logical Eras — Rules for Calculating the Days Elapsed Between Given 
Dates — Equinoctial Time — Fixation of Ancient Dates by Eclipses 

(906.) Time, like distance, may be measured by com* 
parison with standards of any length, and all that is requi- 
site for ascertaining correctly the length of any interval, is 
to be able to apply the standard to the interval throughout 
its whole extent, without overlapping on the one hand, or 
leaving unmeasured vacancies on the other; to determine, 
without the possible error of a unit, the number of integer 
standards which the interval admits of being interposed 
between its beginning and end; and to estimate precisely 
the fraction, over and above an integer, which remains 
when all the possible integers are subtracted. 

(907.) But though all standard units of time are equally 
possible, theoretically speaking, yet all are not, practically, 
equally convenient. The solar day is a natural interval 
which the wants and occupations of man in every state of 
society force upon him, and compel him to adopt as his 
fundamental unit of time. Its length as estimated from the 
departure of the sun from a given meridian, and its next 
return to the same, is subject, it is true, to an annual flue* 



OUTLINES OF ASTRONOMY 819 

tuation in excess and defect of its mean value, amounting 
at its maximum to full half a minute. But except for 
astronomical purposes, this is too small a change to inter- 
fere in the slightest degree with its use, or to attract any 
attention, and the tacit substitution of its mean for its true 
(or variable) value may be considered as having been made 
from the earliest ages, by the ignorance of mankind that 
any such fluctuation existed. 

(908.) The time occupied by one complete rotation of the 
earth on its axis, or the mean 1 sidereal day, may be shown, 
on dynamical principles, to be subject to no variation from 
any external cause, and although its duration would be 
shortened by contraction in the dimensions of the globe 
itself, such as might arise from the gradual escape of its 
internal heat, and consequent refrigeration and shrinking 
of the whole mass, yet theory, on the one hand, has ren- 
dered it almost certain that this cause cannot have effected 
any perceptible amount of change during the history of the 
human race; and, on the other, the comparison of ancient 
and modern observations affords every corroboration to this 
conclusion. From such comparisons, Laplace has concluded 
that the sidereal day has not changed by so much as one hun- 
dredth of a second since the time of Hipparchus. The mean 
sidereal day therefore possesses in perfection the essential 
quality of a standard unit, that of complete invariability. 
The same is true of the mean sidereal year, if estimated upon 
an average sufficiently large to compensate the minute fluctu- 
ations arising from the periodical variations of the major axis 
of the earth's orbit due to planetary perturbation (art. 668). 

(909.) The mean solar day is an immediate derivative of 

1 The true sidereal day is variable by the effect of nutation; but the varia- 
tion (an excessively minute fraction of the whole) compensates itself in a revolu- 
tion of the moon's nodes. 

Astronomy — Vol. XX — 16 



820 



OUTLINES OF ASTRONOMY 



the sidereal day and year, being connected with them by the 
same relation which determines the synodic from the sidereal 
revolutions of any two planets or other revolving bodies 
(art. 418). The exact determination of the ratio of the 
sidereal to the solar day, which is a point of the utmost 
importance in astronomy, is, however, in some degree, 
complicated by the effect of precession, which renders it 
necessary to distinguish between the absolute time of the 
earth's rotation on its axis (the real natural and invariable 
standard of comparison), and the mean interval between two 

successive returns of a given 
star to the same meridian, or 
rather of a given meridian to 
the same star, which not only 
differs by a minute quantity 
from the sidereal day, but is 
actually not the same for all 
stars. As this is a point to 
which a little difficulty of 
conception is apt to attach, it 
will be necessary to explain 
it in some detail. Suppose 
then TV the pole of the ecliptic, and P that of the equinoctial, 
A C the solstitial colure at any given moment of time, and 
P p a r the small circle described by P about n in one revo- 
lution of the equinoxes, i.e. in 25870 years, or 9448300 solar 
days, all projected on the plane of the ecliptic ABCD. 
Let S be a star anywhere situated on the ecliptic, or between 
it and the small circle P q r. Then if the pole P were at 
rest, a meridian of the earth setting out from P S C, and 
revolving in the direction C D, will come again to the star 
after the exact lapse of one sidereal day, or one rotation of 




OUTLINES OF ASTRONOMY 821 

the earth on its axis. But P is not at rest. After the lapse 
of one such day it will have come into the situation (sup- 
pose) p, the vernal equinox B having retreated to 6, and the 
colure P C having taken up the new position^ c. Now 
a conical movement impressed on the axis of rotation of a 
globe already rotating is equivalent to a rotation impressed 
on the whole globe round the axis of the cone, in addition 
to that which the globe has and retains round its own inde- 
pendent axis of revolution. Such a new rotation, in trans- 
ferring P to pj being performed round an axis passing 
through jt, will not alter the situation of that point of the 
globe which has ar in its zenith. Hence it follows that p * c 
passing through ar will be the position taken up by the 
meridian P ic C after the lapse of an exact sidereal day. 
But this does not pass through S, but falls short of it by 
the hour-angle it p S, which is yet to be described before 
the meridian comes up to the star. The meridian, then, has 
lost so much on, or lagged so much behind, the star in the 
lapse of that interval. The same is true whatever be the arc 
Pp. After the lapse of any number of days, the pole being 
transferred to jt>, the spherical angle n p & will measure the 
total hour angle which the meridian has lost on the star. 
Now when S lies anywhere between C and r, this angle 
continually increases (though not uniformly), attaining 180° 
when p comes to r, and still (as will appear by following 
out the movement beyond r) increasing thence till it attains 
360° when p has completed its circuit. Thus in a whole 
revolution of the equinoxes, the meridian will have lost one 
exact revolution upon the star, or in 9448300 sidereal days 
will have reattained the star only 9448299 times: in other 
words, the length of the day measured by the mean of the 
successive arrivals of any star outside of the circle P p q r 



822 OUTLINES OF ASTRONOMY 

on one and the same meridian is to the absolute time of 
rotation of the earth on its axis as 9448300 : 9448299, 
or as 1-00000011 to 1. 

(910.) It is otherwise of a star situated within this circle, 
as at <t. For such a star the angle * p <?, expressing the lag- 
ging of the meridian, increases to a maximum for some 
situation of p between q and r, and decreases again to o at 
r\ after which it takes an opposite direction, and the merid- 
ian begins to get in advance of the star, and continues to 
get more and more so, till p has attained some point between 
s and P, where the advance is a maximum, and thence de- 
creases again to o when p has completed its circuit. For 
any star so situated, then, the mean of all the days so esti- 
mated through a whole period of the equinoxes is an abso- 
lute sidereal day, as if precession had no existence. 

(911.) If we compare the sun with a star situated in the 
ecliptic, the sidereal year is the mean of all the intervals of 
its arrival at that star throughout indefinite ages, or (without 
fear of sensible error) throughout recorded history. Now, 
if we would calculate the synodic sidereal revolution of the 
sun and of a meridian of the earth by reference to a star so 
situated, according to the principles of art. 418, we must 
proceed as follows: Let D be the length of the mean solar 
day (or synodic day in question), d the mean sidereal rev- 
olution of the meridian with reference to the same star, 
and y the sidereal year. Then the arcs described by the 
sun and the meridian in the interval D will be respectively 

300° — and 360° — . And since the latter of these exceeds 
y d 

the former by precisely 360°, we have 

360° ^=360° - +360°': 
d y 



OUTLINES OF ASTRONOMY 823 

whence it follows that 

5 = l + - = 1-00273780, 
d y 

taking the valae of the sidereal year y as given in art. 383, 
viz. 865 d 6 h 9 m 9*6 8 . But, as we have seen, d is not the abso- 
lute sidereal day, but exceeds it in the ratio 1-00000011 : 1. 
Hence to get the value of the mean solar day, as expressed 
in absolute sidereal days, the number above set down must 
be increased in the same ratio, which brings it to 1*00273791, 
which is the ratio of the solar to the sidereal day actually in 
use among astronomers. 

(912.) It would be well for chronology if mankind 
would, or could have contented themselves with this one 
invariable, natural, and convenient standard in their reckon- 
ing of time. The ancient Egyptians did so, and by their 
adoption of a historical and official year of 365 days have 
afforded the only example of a practical chronology, free 
from all obscurity or complication. But the return of the 
seasons, on which depend all the more important arrange- 
ments and business of cultivated life, is not conformable to 
such a multiple of the diurnal unit. Their return is regu- 
lated by the tropical year, or the interval between two suc- 
cessive arrivals of the sun at the vernal equinox, which, as 
we have seen (art. 383), differs from the sidereal year by 
reason of the motion of the equinoctial points. Now this 
motion is not absolutely unifom, because the ecliptic, upon 
which it is estimated, is gradually, though very slowly, 
changing its situation in space under the disturbing influ- 
ence of the planets (art. 640). And thus arises a variation 
in the tropical year, which is dependent on the place of the 
equinox (art 383). The tropical year is actually about 4-21 3 



824 OUTLINES OF ASTRONOMY 

shorter than it was in the time of Hipparchus. This ab- 
sence of the most essential requisite for a standard, viz. in- 
variability, renders it necessary, since we cannot help em- 
ploying the tropical year in our reckoning of time, to adopt 
an arbitrary or artificial value for it, so near the truth as 
not to admit of the accumulation of its error for several cen- 
turies producing any practical mischief, and thus satisfying 
the ordinary wants of civil life; while, for scientific pur- 
poses, the tropical year, so adopted, is considered only as the 
representative of a certain number of integer clays and a 
fraction — the day being, in effect, the only standard em- 
ployed. The case is nearly analogous to the reckoning of 
value by guineas and shillings, an artificial relation of the 
two coins being fixed by law, near to, but scarcely ever ex- 
actly coincident with, the natural one, determined by the 
relative market price of gold and silver, of which either the 
one or the other — whichever is really the most invariable, 
or the most in use with other nations — may be assumed as 
the true theoretical standard of value. 

(913.) The other inconvenience of the tropical year as a 
greater unit is its incommensurability with the lesser. In 
our measure of space all our subdivisions are into aliquot 
parts: a yard is three feet, a mile eight furlongs, etc. But 
a year is no exact number of days, nor an integer number 
with any exact fraction, as one-third or one-fourth, over and 
above; but the surplus is an incommensurable fraction, com- 
posed of hours, minutes, seconds, etc., which produces the 
same kind of inconvenience in the reckoning of time that it 
would do in that of money, if we had gold coins of the 
value of twenty-one shillings, with odd pence and farthings, 
and a fraction of a farthing over. For this, however, there 
is no remedy but to keep a strict register of the surplus 



OUTLINES OF ASTRONOMY 825 

fractions: and, when they amount to a whole day, cast them 
over into the integer account. 

(914.) To do this in the simplest and most convenient 
manner is the object of a well-adjusted calendar. In the 
Gregorian calendar, which we follow, it is accomplished with 
considerable simplicity and neatness, by carrying a little fur- 
ther than is done above, the principle of an assumed or arti- 
ficial year, and adopting two such years, both consisting of 
an exact integer number of days, viz. one of 365 and the 
other of 366, and laying down a simple and easily remem- 
bered rule for the order in which these years shall succeed 
each other in the civil reckoning of time, so that during the 
lapse of at least some thousands of years the sum of the in- 
teger artificial, or Gregorian, years elapsed shall not differ 
from the same number of real tropical years by a whole day. 
By this contrivance, the equinoxes and solstices will always 
fail on days similarly situated, and bearing the same name 
in each Gregorian year; and the seasons will forever corre- 
spond to the same months, instead of running the round of 
the whole year, as they must do upon any other system of 
reckoning, and used, in fact, to do before this was adopted, 
as a matter of ignorant haphazard in the Greek and Eoman 
chronology, and of strictly defined and superstitiously rigor- 
ous observance in the Egyptian. 

(915.) The Gregorian rule is as follows: — The years are 
denominated as years current {not as years elapsed) from the 
midnight between the 31st of December and the 1st of Janu- 
ary immediately subsequent to the birth of Christ, accord- 
ing to the chronological determination of that event by 
Dionysius Exiguus. Every year whose number is not di- 
visible by 4 without remainder, consists of 365 days; every 
year which is so divisible, but is not divisible by 100, of 



826 OUTLINES OF ASTRONOMY 

366; every year divisible by 100, but not by 400, again of 
365; and every year divisible by 400, again of 366. For 
example, the year 1833, not being divisible by 4, consists of 
365 days; 1836 of 366; 1800 and 1900 of 365 each; but 2000 
of 366. In order to see how near this rule will bring us to 
the truth, let us see what number of days 10000 Gregorian 
years will contain, beginning with the year A.D. 1. Now, 
in 10000, the numbers not divisible by 4 will be | of 10000 
or 7500; those divisible by 100, but not by 400, will in like 
manner be § of 100, or 75; so that, in the 10000 years in 
question, 7575 consist of 365, and the remaining 2425 of 366, 
producing in all 3652425 days, which would give for an 
average of each year, one with another, 365 d *2425. The 
actual value of the tropical year (art. 383), reduced into a 
decimal fraction, is 365*24224, so the error in the Gregorian 
rule on 10000 of the present tropical years, is 2*6, or 2 d 14 h 
24 m ; that is to say, less than a day in 3000 years ; which is 
more than sufficient for all human purposes, those of the 
astronomer excepted, who is in no danger of being led into 
error from this cause. Eveu this error is avoided by ex- 
tending the wording of the Gregorian rule one step further 
than its contrivers probably thought it worth while to go, 
and declaring that years divisible by 4000 should consist of 
365 days. This would take or! two integer days from the 
above calculated number, and 2*5 from a larger average; 
making the sum of days in 100000 Gregorian years, 36524225, 
which differs only by a single day from 100000 real tropical 
years, such as they exist at present. 2 

(916.) In the historical dating of events there is no year 
A.D. 0. The year immediately previous to A.D. 1 is al- 
ways called B.C. 1. This must always be borne in mind in 

8 See note A at the end of the chapter. 



OUTLINES OF ASTRONOMY 827 

reckoning chronological and astronomical intervals. The 
sum of the nominal } T ears B.C. and A.D. must be dimin- 
ished by 1. Thus, from January 1, B.C. 4713, to January 
1. A.D. 1582, the years elapsed are not 6295, but 6294. 

(917.) As any distance along a highroad might, though 
in a rather inconvenient and roundabout way, be expressed 
without introducing error by setting up a series of mile- 
stones, at intervals of unequal lengths, so that every fourth 
mile, for instance, should be a yard longer than the rest, or 
according to any other fixed rule; taking care only to mark 
the stones so as to leave room for no mistake, and to adver- 
tise all travellers of the difference of lengths and their order 
of succession; so may any interval of time be expressed 
correctly by stating in what Gregorian years it begins and 
ends, and whereahout in each. For this statement coupled 
with the declaratory rule, enables us to say how many in- 
teger years are to be reckoned at 365, and how many at 366 
days. The latter years are called bissextiles, or leap-years, 
and the surplus days thus thrown into the reckoning are 
called intercalary or leap-days. 

(918.) If the Gregorian rule, as above stated, had always 
and in all countries been known and followed, nothing 
would be easier than to reckon the number of days elapsed 
between the present time, and any historical recorded event. 
But this is not the case; and the history of the calendar, 
with reference to chronology, or to the calculation of ancient 
observations, may be compared to that of a clock, going 
regularly when left to itself, but sometimes forgotten to be 
wound up; and when wound, sometimes set forward, some- 
times backward, either to serve particular purposes and pri- 
vate interests, or to rectify blunders in setting. Such, at 
least, appears to have been the case with the Eoman calen- 



828 OUTLINES OF ASTRONOMY 

dar, in which our own originates, from the time of Numa to 
that of Julius Caesar, when the lunar year of 13 months, or 
355 days, was augmented at pleasure to correspond to the 
solar, by which the seasons are determined, by the arbitrary 
intercalations of the priests, and the usurpations of the de- 
cemvirs and other magistrates, till the confusion became in- 
extricable. To Julius Caesar, assisted by Sosigenes, an emi- 
nent Alexandrian astronomer and mathematician, we owe 
the neat contrivance of the two years of 365 and 366 days, 
and the insertion of one bissextile after three common years. 
This important change took place in the 45th year before 
Christ, which he ordered to commence on the 1st of January, 
being the day of the new moon immediately following the winter 
solstice of the year before. We may judge of the state into 
which the reckoning of time had fallen, by the fact, that to in- 
troduce the new system it was necessary to enact that the pre- 
vious year, 46 B.C., should consist of 445 days, a circumstance 
which obtained for it the epithet of "the year of confusion." 
(919.) Had Caesar lived to carry out into practical effect, 
as Chief Pontiff, his own reformation, an inconvenience 
would have been avoided, which at the very outset threw 
the whole matter into confusion. The words of his edict 
establishing the Julian system have not been handed down 
to us, but it is probable that they contained some expression 
equivalent to "every fourth year," which the priests misin- 
terpreting after his death to mean (according to the sacer- 
dotal system of numeration) as counting the leap year newly 
elapsed as No. 1, of the four, intercalated every third instead 
of every 4th year. This erroneous practice continued dur- 
ing 36 years, in which therefore 12 instead of 9 days were 
intercalated, and an error of three days produced; to rectify 
which r Augustus ordered the suspension of all intercalation 



OUTLINES OF ASTRONOMY 829 

during three complete quadriennia — thus restoring, as may 
be presumed his intention to have been, the Julian dates for 
the future, and re-establishing the Julian system, which was 
never afterward vitiated by any error, till the epoch when 
its own inherent defects gave occasion to the Gregorian ref- 
ormation. According to the Augustan reform the years 
A.U.C. 761, 765, 769, etc., which we now call A.D. 8, 12, 
16, etc., are leap years. And starting from this as a certain 
fact (for the statements of the transaction by classical authors 
are not so precise as to leave absolutely no doubt as to the 
previous intermediate years), astronomers and chronologists 
have agreed to reckon backward in unbroken succession on 
this principle, and thus to carry the Julian chronology into 
past time, as if it had never suffered such interruption, and 
as if it were certain 3 that Caesar, by way of securing the in- 
tercalation as a matter of precedent, made his initial year 45 
B.C. a leap year. Whenever, therefore, in the relation of 
any event, either in ancient history, or in modern, previous 
to the change of style, the time is specified in our modern 
nomenclature, it is always to be understood as having been 
identified with the assigned date by threading the mazes 
(often very tangled and obscure ones, of special and na- 
tional chronology, and referring the day of its occurrence 
to its place in the Julian system so interpreted. 

(920.) Different nations in different ages of the world 
have of course reckoned their time in different ways, and 

3 Scaliger, Ideler, and the best authorities consider it probable. A strong, 
if not decisive, argument in its favor, is that Augustus evidently intending to 
reinstate the Julian idea, and with a clear view of the recent inconveniences 
present to his mind, did actually direct the future intercalations to take place in 
odd years U.C. Such then, no doubt, must have been Caesar's intention. For 
the correction of Roman dates during the fifty-two years between the Julian 
and Augustan reformations, see Ideler, "Handbuch der Mathematischen und 
Technischen Chronologie, " which we take for our guide throughout this 
chapter. 



830 OUTLINES OF ASTRONOMY 

from different epochs, and it is therefore a matter of great 
convenience that astronomers and chronologists (as they 
have agreed on the uniform adoption of the Julian system 
of years and months) should also agree on an epoch antece- 
dent to them all r to which, as to a fixed point in time, the 
whole list of chronological eras can be differentially referred. 
Such an epoch is the noon* of the 1st of January, B.C. 4.713, 
which is called the epoch of the Julian period, a cycle of 
7980 Julian years, to understand the origin of which, we 
must explain that of three subordinate cycles, from whose 
combination it takes its rise, by the multiplication together 
of the numbers of years severally contained in them; viz. 
the Solar and Lunar cycles, and that of the indictions. 

(921.) The Solar cycle consists of 28 Julian years, after 
the lapse of which the same days of the week on the Julian 
system would always return to the same days of each month 
throughout the year. For four such years consisting of 1461 
days, which is not a multiple of 7, it is evident that the 
least number of years which will fulfil this condition must 
be seven times that interval, or 28 years. The place in this 
cycle for any year A.D., as 1849, is found by adding 9 to 
the year, and dividing by 28. The remainder is the number 
sought, being counted as 28. 

(922.) The Lunar cycle consists of 19 years or 235 luna- 
tions, which differ from 19 Julian years of 365| days only 
by about an hour and a half, so that, supposing the new 
moon to happen on the first of January, in the first year of 
the cycle, it will happen on that day (or within a very short 
time of its beginning or ending) again after a lapse of 19 
years; and all the new moons in the interval will run on the 
same days of the month as in the preceding cycle. This 
period of 19 years is sometimes called the Metonic cycle, 



OUTLINES OF ASTRONOMY 831 

from its discoverer Meton, an Athenian mathematician, a 
discovery duly appreciated by his countrymen, as insuring 
the correspondence between the lunar and solar years, the 
former of which was followed by the Greeks. Public hon- 
ors were decreed to him for this discovery, a circumstance 
very expressive of the annoyance which a lunar year of 
necessity inflicts on a civilized people, to whom a regular 
and simple calendar is one of the first necessities of life. A 
cycle of 4x19=76 years was proposed by Callippus as a 
supposed improvement on the Metonic, but in this interval 
the errors accumulated to 6 hours and in 4x76=304 years 
to an entire day. To find the place of a given year in the 
lunar cycle (or as it is called the Golden Number), add 1 to 
the number of the year A.D., and divide by 19, the re- 
mainder (or 19 if exactly divisible), is the Golden Number. 

(923.) The cycle of the indictions is a period of 15 years 
used in the courts of law, and in the fiscal organization of 
the Eoman empire, under Constantine and his successors, 
and thence introduced into legal dates, as the Golden Num- 
ber, serving to determine Easter, was into ecclesiastical 
ones. To find the place of a year in the indiction cycle, 
add 3 and divide by 15. The remainder (or 15 if remain) 
is the number of the inclictional year. 

(924.) If we multiply together the numbers 28, 19, and 
15, we get 7980, and, therefore, a period or cycle of 7980 
years will bring round the years of the three cycles again in 
the same order, so that each year shall hold the same place 
in all the three cycles as the corresponding year in the fore- 
going period. As none of the three numbers in question 
have any common factor, it is evident that no two years in 
the same compound period can agree in all the three par- 
ticulars: so that to specify the numbers of a year in each of 



832 OUTLINES OF ASTRONOMY 

these cycles is, in fact, to specify the year, if within that 
long period; which embraces the entire of authentic chro- 
nology. The period thus arising of 7980 Julian years, is 
called the Julian period, and it has been found so useful, 
that the most competent authorities have not hesitated to 
declare that, through its employment, light and order were 
first introduced into chronology. 4 We owe its invention or 
revival to Joseph Scaliger, who is said to have received it 
from the Greeks of Constantinople. The first year of the 
current Julian period, or that of which the number in each 
of the three subordinate cycles is 1, was the year 4713 B.C., 
and the noon of the 1st of January of that year, for the 
meridian of Alexandria, is the chronological epoch, to which 
all historical eras are most readily and intelligibly referred, 
by computing the number of integer days intervening be- 
tween that epoch and the noon (for Alexandria) of the day, 
which is reckoned to be the first of the particular era in 
question. The meridian of Alexandria is chosen as that to 
which Ptolemy refers the commencement of the era of Na- 
bonassar, the basis of all his calculations. 

(925.) Given the year of the Julian period, those of the 
subordinate cycles are easily determined as above. Con- 
versely, given the years of the solar and lunar cycles and of 
the indiction, to determine the year of the Julian period 
proceed as follows: — Multiply the number of the year in 
the solar cycle by 4845, in the lunar by 4200, and in the 
Cycle of the Indictions by 6916, divide the sum of the prod- 
ucts by 7980, and the remainder is the year of the Julian 
period sought. 

(926.) The following table contains these intervals for 
some of the more important historical eras: — 

4 Ideler, Handbucb, etc., vol. i. p. 37. 



OUTLINES OF ASTRONOMY 



833 



Intervals in Days between the Commencement of the Julian Period, and that of 
some other remarkable chronological and astronomical Eras 5 







Chron- 


Cur- 


Interval, 


Names by which the Era is usually cited 


First day 

current of 

the Era 


ologi- 
cal Des- 
igna- 
tion of 
the 
Year 


rent 
Year of 

the 
Julian 


days 
elapsed. 
For days 
current 






Period 


addl 


Jvlian Epochs 


Julian Dates 


B.C. 






Julian period 


Jan. 1. 


4713 


1 





Creation of the world (Usher) . 


(Jan. 1.) 


4004 


710 


258,963 


Era of the Deluge (Aboulhassan Ku- 










schiar) 


Feb. 18. 


3102 


1612 


588,466 


Ditto Vulgar Computation . 


(Jan. 1.) 


2348 


2366 


863,817 


Era of Abraham (Sir H. Nicolas) 


Oct. 1. 


2015 


2699 


985,718 


Destruction of Troy, (ditto) 


July 12. 


1184 


3530 


1,289,160 


Dedication of Solomon's Temple 


(May 1.) 


1015 


3699 


1,350,815 


Olympiads (mean epoch in general use) 


July 1. 


m 


3938 


1,438,171 


Building of Rome (Varronian epoch, 










U.C.) 


April 22. 


753 


3961 


1,446,502 


Era of Nabonassar . 


Feb. 26. 


747 


3967 


1,448,638 


Eclipse of Thales . 


May 28. 


585 


4129 


1,507,900 


Eclipse of Larissa . 


May 19. 


557 


4157 


1,518,118 


Me tonic cycle (Astronomical epoch) . 


July 15. 


432 


4282 


1,563,831 


Callippic cycle Do. (Biot) 


June 28. 


330 


4384 


1,599,608 


Philippic era, or era of Philip Aridaeus 


Nov. 12. 


324 


4390 


1,603,398 


Era of the Seleucidse . 


Oct. 1. 


312 


4402 


1,607,739" 


Eclipse of Agathocles 


Aug. 15. 


310 


4404 


1,608,422 


Cesarean era of Antioch . 


Sept. 1. 


49 


4665 


1,703,770 


Julian reformation of the Calendar 


Jan. 1. 


45 


4669 


1,704,987 


Spanish era 


Jan. 1. 


38 


4676 


1,707,544 


Actian era in Rome . 


Jan. 1. 


30 


4684 


1,710,466 


Actian era of Alexandria . 


Aug. 29. 


30 


4684 


1,710,706 


Vulgar or Dionysian era 


Jan. 1. 


A.D. 1 


4714 


1,721,424 


Era of Diocletian . 


Aug. 29. 


284 


4997 


1,825,030 


Hejira (astronomical epoch, new moon) 


July 15. 


622 


5335 


1,948,439 


Era of Yezdegird . 


June 16. 


632 


5345 


1,952,063 


Eclipse of Sticklastad 


Aug. 31. 


1030 


5743 


2,097,508 


Gelalsean era (Sir H. Nicolas) 


March 14. 


1079 


5792 


2,115,235 


Last day of Old Style (Catholic nations] 


Oct. 4. 


1582 


6295 


2,299,160 


Last day of Old Style in England 


Sept. 2. 


1752 


6465 


2,361,221 


Gregorian Epochs 


Gregorian 
Dates 








New Style in Catholic nations . 


Oct. 15. 


1582 


6295 


2,299,161 


Ditto in England 


Sept. 14. 


1752 


6465 


2,361,222 


Commencement of the 19th century. 










Epoch of Bode's catalogue of stars. 


Jan. 1. 


1801 


6514 


2,378,862 


Epoch of the catalogue of stars of the 










R. Astronomical Society 


Jan. 1. 


1830 


6543 


2,389,454 


Epoch of the catalogue of the British 










Association 


Jan. 1. 


1850 


6563 


2,396,759 



6 See note B at the end of this chapter. 



834 OUTLINES OF ASTRONOMY 

(927.) The determination of the exact interval between 
any two given dates, is a matter of such importance, and, 
unless methodically performed, is so very liable to error, 
that the following rules will not be found out of place. In 
the first place it must be remarked, generally, that a date, 
whether of a day or year, always expresses the day or year 
current and not elapsed, and that the designation of a year 
by A.D. or B.C. is to be regarded as the name of that year, 
and not as a mere number uninterruptedly designating the 
place of the year in the scale of time. Thus in the date, 
Jan. 5, B.C. 1, Jan. 5 does not mean that 5 days of Jan- 
uary in the year in question have elapsed, but that 4 have 
elapsed, and the 5th is current. And the B.C. 1, indicates 
that the first day of the year so named (the first year current 
before Christ), preceded the first day of the vulgar era by 
one year. The scale of A.D. and B.C., as already ob- 
served, is not continuous, the year in both being wanting; 
so that (supposing the vulgar reckoning correct) our Saviour 
was born in the year B.C. 1. 

(928.) To find the year current of the Julian period (J. P.) 
corresponding to any given year current B.C. or A.D. If 
B.C., subtract the number of the year from 4714; if A.D., 
add its number to 4713. For examples, see the foregoing 
table. 

(929.) To find the day current of the Julian period corre- 
sponding to any given date, Old Style. Convert the year 
B.C. or A.D. into the corresponding year J. P. as above. 
Subtract 1 and divide the number so diminished by 4, and 
call Q the integer quotient, and E the remainder. Then 
will Q be the number of entire guadriennia of 1461 days 
each, and P the residual years, the first of which is always 
a leap year. Convert Q into days by the help of the first 



OUTLINES OF ASTRONOMY 



835 



of the annexed tables, and R by the second, and the sum 
will be the interval between the Julian epoch, and the com- 
mencement, January 1, of the year. Then find the days 
intervening between the beginning of January 1, and that 
of the date-day by the third table, using the column for a 
leap year, where R=0, and that for a common year when 
R is 1, 2 or 3. Add the days so found to those in Q+R? 
and the sum will be, the days elapsed of the Julian period, 
the number of which increased by 1 gives the day current. 



Table 1.— Mull 


iples of 1461, the days in a Julian 
Quadriennium 


1 
2 
3 


1461 
2922 

4383 


4 
5 
6 


5844 
7305 
8766 


7 
8 
9 


10227 
11688 
13149 



Table 2.— Days in 


Residual years 








1 


366 


2 


731 


3 


1096 



Table 3.— Days elapsed from Jan. 1, to the 1st of each Month 




In a common 


In a leap 




In a common 


In a leap 




Year 


Year 




Year 


Year 


Jan. 1 . . 








July 1 . . 


181 


182 


Feb. 1 . . 


31 


31 


Aug. 1 . . 


212 


213 


March 1 


59 


60 


Sept. 1 . 


243 


244 


April 1 . . 


90 


91 


Oct. 1 . . 


273 


274 


May 1 . 


120 


121 


Xov. 1 . . 


304 


305 


June 1 . 


151 


152 


Dec. 1 . . 


334 


335 



Example. — What is the current day of the Julian period 
corresponding to the last day of Old Style in England, on 
September 2, A.D. 1752. 



1752 
4713 

6465 year current. 
1 * 
4)6461 years elapsed. 
Q=1616 
B= 



Jan. 
Sept. 1 



1000 




1,461,000 


600 




876,600 


10 




14,610 


6 




8,766 


R=0 







1. to Sept. 


I. 


244 


1. to Sept. 


2, 


1 
2,361,221 days 


Current day 


the 2,361, 222 d . 



836 OUTLINES OF ASTR0N03IY 

(930.) To find the same for any given date, New Style. 
Proceed as above, considering the date as a Julian date, 
and disregarding the change of style. Then from the 
resulting days, subtract as follows: — 

For any date of New Style, antecedent to March 1, A.D. 1700 . 10 days. 

After Feb. 28, 1700 and before March 1, A.D. 1800 .... 11 days. 

1800 " " 1900 .... 12 days. 

1900 " " 2100 .... 13 days, etc. 

(931.) To find the interval between any two dates, whether' 
of Old or New Style, or one of one, and one of the other. 
Find the day current of the Julian period corresponding 
to each date, and their difference is the interval required. 
If the dates contain hours, minutes and seconds, they must 
be annexed to their respective days current, and the sub- 
traction performed as usual. 

(932.) The Julian rule made every fourth year, without 
exception, a bissextile, beginning with the year J. P. 1, 
which is to be accounted as such. This is, in fact, an over- 
correction; it supposes the length of the tropical year to 
be 365 4 d , which is too great, and thereby induces an error 
of 7 days in 900 years, as will easily appear on trial. Ac- 
cordingly, so early as the year 1414, it began to be per- 
ceived that the equinoxes were gradually creeping away 
from the 21st of March and September, where they ought 
to have always fallen had the Julian year been exact, and 
happening (as it appeared) too early. The necessity of a 
fresh and effectual reform in the calendar was from that time 
continually urged, and at length admitted. The change 
(which took place under the popedom of Gregory XIII.) 
consisted in the omission of ten 6 nominal days after the 
4th of October, 1582 (so that the next day was called the 

6 See note C at the end of this chapter. 



OUTLIXES OF ASTRO XO MY 837 

15th, and not the 5th), and the promulgation of the rule 
already explained for future regulation. The change was 
adopted immediately in all Catholic countries; but more 
slowly in Protestant. In England, "the change of style," 
as it was called, took place after the 2d of September, 1752, 
eleven nominal days being then struck out; so that, the last 
day of Old Style being the 2d, the first of New Style (the 
next day) was called the 14th, instead of the 3d. The same 
legislative enactment which established the Gregorian year 
in England in 1752, shortened the preceding year, 1751, by 
a full quarter. Previous to that time, the year was held to 
begin with the 25th March, and the year A.D. 1751 did so 
accordingly; but that year was not suffered to run out, but 
was supplanted on the 1st January by the year 1752, which 
(as well as every subsequent year) it was enacted should 
commence on that day, so that our English year 1751 was 
in effect an "annus confusionis," and consisted of only 282 
days. Russia is now the only country in Europe in which 
the Old Style is still adhered to, and (another secular year 
having elapsed) the difference between the European and 
Eussian dates amounts, at present, to 12 days. 

(933.) It is fortunate for astronomy that the confusion of 
dates, and the irreconcilable contradictions which historical 
statements too often exhibit, when confronted with the best 
knowledge we possess of the ancient reckonings of time, 
affect recorded observations but little. An astronomical 
observation, of any striking and well-marked phenomenon, 
carries with it, in most cases, abundant means of recovering 
its exact date, when any tolerable approximation is afforded 
to it by chronological records; and, so far from being ab- 
jectly dependent on the obscure and often contradictory 
dates, which the comparison of ancient authorities indi- 



838 OUTLINES OF ASTRONOMY 

cates, is often itself the surest and most convincing evi- 
dence on which a chronological epoch can be brought 
to rest. Eemarkable eclipses, for instance, now that the 
lunar theory is thoroughly understood, can be calculated 
back for several thousands of years, without the possi- 
bility of mistaking the day of their occurrence. And, 
whenever any such eclipse is so interwoven with the ac- 
count given by an ancient author of some historical event, 
as to indicate precisely the interval of time between the 
eclipse and the event, and at the same time completely to 
identify the eclipse, that date is recovered and fixed for- 
ever. This has been done in the cases of four very remark- 
able total eclipses of the sun (the dates of which are accord- 
ingly entered as epochs in the table of Chronological Eras, 
art. 926), which have given rise (at least one of them) to 
much discussion and diversity of opinion among astrono- 
mers, but which have at length been definitively settled by 
Mr. Airy on the occasion of the recent publication of Prof. 
Hansen's Lunar Tables, the accuracy of which is such as 
to justify the most entire reliance on the results of such 
calculations grounded upon them. 

(933 a.) The solar eclipse designated as that "of Thales" 
is the celebrated one which is stated by Herodotus to have 
been predicted by that philosopher, and to have caused the 
suspension of a battle between the Medes and Lydians, fol- 
lowed by a treaty of peace. Only a total eclipse, as Mr. 
Baily had clearly shown, could have so attracted the atten- 
tion of the combatants; and in a very remarkable memoir 
on the subject (Phil. Trans. 1811) that eminent astronomer 
was led, by the use of the best tables then in existence, to 
identify this eclipse with the total one of September 30th, 
B.C. 610, which, according to those tables, must have 



OUTLINES OF ASTRONOMY 839 

passed over the mouth of the Eiver Halys (where it 
had all along been assumed, though without any positive 
grounds for the assumption, the battle was fought). The 
same conclusion having been arrived at by M. Oltmanns, 
the point was supposed to be settled. Prof. Hansen's ta- 
bles, however, throw the path of the shadow in this eclipse 
altogether out of Asia Minor, and even north of the Sea of 
Azof. On the other hand the eclipse of B.C. 585, which 
was also total, passed, according to those tables, over Issus, 
a locality satisfying all the circumstantial and general mili- 
tary conditions of the narrative even better than the Halys, 
and at this spot there can now be little or no doubt the 
battle was really fought, 

(933 b.) The total eclipse of the sun which was witnessed 
by the fleet of Agathocles in his escape from Syracuse, 
blockaded by the Carthaginians, on the second day of his 
voyage to Cape Bon, had been considered by Mr. Baily 
in the memoir above cited, and found to be incompatible 
(according to the then existing tables) with the year B.C. 
310, supposing the former eclipse to have been rightly identi- 
fied. This having now been shown not to be the case, it 
is all the more satisfactory to find that, under very reason- 
able and natural suppositions respecting Agathocles' voy- 
age, the total eclipse which did undoubtedly pass on the 
date assigned very near the southern corner of Sicily might 
have enveloped his fleet, and that no other eclipse by possi- 
bility could have done so. 

(933 c.) The "eclipse of Larissa" is related by Xeno- 
phon to have caused the capture of the Asiatic city of that 
name, by producing a panic terror in its Median defenders, 
of which the Persian besiegers took advantage. The site 
of Larissa has been satisfactorily identified by Mr. Layard 



840 OUTLINES OF ASTRONOMY 

with Nimkoud, and a pyramid of stone close to it (the city 
being of burned brick on a substructure of stone) is ex- 
pressly mentioned by Xenophon, the remains of which still 
exist; as well as those of a neighboring castle near Mespila 
(Mosul) built of shelly stone (also expressly mentioned as 
such by the Greek historian). According to Hansen's ta- 
bles the total eclipse of May 19th, B.C. 557, passed centrally 
over Nimroud, and — the total shadow being in this instance 
a very small one, not exceeding some 25 miles in diameter 
— we are thus presented with a datum, in those remote 
times, having all the precision of a most careful modern 
observation, not only for establishing a chronological epoch, 
but for affording a point of reference in the history of 
the moon's motion. The eclipse of Sticklastad is of no his- 
torical importance. 7 

(984.) The days thus parcelled out into years, the next 
step to a perfect knowledge of time is to secure the identifi- 
cation of each day, by imposing on it a name universally 
known and employed. Since, however, the days of a whole 
year are too numerous to admit of loading the memory with 
distinct names for each, all nations have felt the necessity 
of breaking them down into parcels of a more moderate 
extent; giving names to each of these parcels, and particu- 
larizing the days in each by numbers, or by some especial 
indication. The lunar month has been resorted to in many 
instances ; and some nations have, in fact, preferred a lunar 
to a solar chronology altogether, as the Turks and Jews 
continue to do to this day, making the year consist of 12 
lunar months, or 354 days. Oar own division into twelve 

* The solar eclipse in the first year of the Peloponnesian War, which was 
total at Athens, "some stars becoming visible," according to Thuej T dides, de- 
serves to be recomputed. See Heis. He supposes the eclipse not total, and 
the "stars" to have been planets. 



OUTLINES OF ASTRONOMY 841 

unequal months is entirely arbitrary, and often productive 
of confusion, owing to the equivoque between the lunar 
and calendar month. 8 The intercalary day naturally at- 
taches itself to February as the shortest month. 

(935.) Astronomical time reckons from the noon of the 
current day; civil from the preceding midnight, so that 
the two dates coincide only during the earlier half of the 
astronomical and the latter of the civil day. This is an 
inconvenience which might be remedied by shifting the 
astronomical epoch to coincidence with the civil. There is, 
however, another inconvenience, and a very serious one, 
to which, both are liable, inherent in the nature of the day 
itself, which is a local phenomenon, and commences at 
different instants of absolute time, under different me- 
ridians, whether we reckon from noon, midnight, sunrise, 
or sunset. In consequence, all astronomical observations 
require in addition to their date, to render them compar- 
able with each other, the longitude of the place of observa- 
tion from some meridian, commonly respected by all astron- 
omers. For geographical longitudes, the Isle of Ferroe has 
been chosen by some as a common meridian, indifferent 
(and on that very account offensive) to all nations. Were 
astronomers to follow such an example, they would prob- 
ably fix upon Alexandria, as that to which Ptolemy's ob- 
servations and computations were reduced, and as claiming 
on that account the respect of all while offending the 
national egotism of none. But even this will not meet 
the whole difficulty. It will still remain doubtful, on a 
meridian of 180° remote from that of Alexandria, what 

8 "A month in law is a lunar month or twenty-eight days" (!! see Art. 
418), "unless otherwise expressed." — Blackstone, ii. chap. 9: "a lease for 
twelvemonths is only for forty-eight weeks." — Ibid.; yet the same eminent 
authority (Introd. § 3) informs us that "the law is the perfection of reason," 
and that "what is not reason is not law." 



842 OUTLINES OF ASTRONOMY 

day is intended by any given date. Do what we will, whea 
it is Monday the 1st of January, 1849, in one part of the 
world, it will be Sunday the 31st of December, 1848, in 
another, so long as time is reckoned by local hours. This 
equivoque, and the necessity of specifying the geographical 
locality as an element of the date, can only be got over by 
a reckoning of time which refers itself to some event, real 
or imaginary, common to all the globe. Such an event is 
the passage of the sun through the vernal equinox, or rather 
the passage of an imaginary sun, supposed to move with 
perfect equality, through a vernal equinox supposed free 
from the inequalities of nutation, and receding upon the 
ecliptic with perfect uniformity. The actual equinox is 
variable, not only by the effect of nutation, but by that 
of the inequality of precession resulting from the change 
in the plane of the ecliptic due to planetary perturbation. 
Both variations are, however, periodical, the one, in the 
short period of 19 years, the other, in a period of enormous 
length, hitherto, uncalculated, and whose maximum of 
fl actuation is also unknown. This would appear, at first 
sight, to render impracticable the attempt to obtain from 
the sun's motion any rigorously uniform measure of time. 
A little consideration, however, will satisfy us that such 
is not the case. The solar tables, by which the apparent 
place of the sun in the heavens is represented with almost 
absolute precision from the earliest ages to the present 
time, are constructed upon the supposition that a certain 
angle, which is called "the sun's mean longitude" (and 
which is in effect the sum of the mean sidereal motion of 
the sun, plus the mean sidereal motion of the equinox in 
the opposite direction, as near as it can be obtained from 
the accumulated observations of twenty-five centuries), in- 



OUTLINES OF ASTRONOMY 843 

creases with rigorous uniformity as time advances. The 
conversion of this mean longitude into time at the rate of 
360° to the mean tropical year (such as the tables assume 
it), will therefore give us both the unit of time, and the 
uniform measure of its lapse, which we seek. It will also 
furnish us with an epoch, not indeed marked by any real 
event, but not on that account the less positively fixed, 
being connected, through the medium of the tables, with 
every single observation of the sun on which they have 
been constructed and with which compared. 

(936.) Such is the simplest abstract conception of equi- 
noctial time. It is the mean longitude of the sum of some 
one approved set of solar tables, converted into time at the 
rate of 360° to the tropical year. Its unit is the mean 
tropical year which those tables assume, and no other, 
and its epoch is the mean vernal equinox of these tables 
for the current year, or the instant when the mean longi- 
tude of the tables is rigorously 0, according to the assumed 
mean motion of the sun and equinox, the assumed epoch 
of mean longitude, and the assumed equinoctial point on 
which the tables have been computed, and no other. To 
give complete effect to this idea, it only remains to specify 
the particular tables fixed upon for the purpose, which 
ought to be of great and admitted excellence, since, once 
decided on, the very essence of the conception is that no 
subsequent alteration in any respect should be made, even 
when the continual progress of astronomical science shall have 
shown any one or all of the elements concerned to be in some 
minute degree erroneous (as necessarily they must), and shall 
have even ascertained the corrections they require (to be them- 
selves again corrected, when another step in refinement 

shall have been made). 

Astronomy — Vol. XX — 17 



844 OUTLINES OF ASTRONOMY 

(937.) Delambre's solar tables (in 1828), when this mode 
of reckoning time was first introduced, 9 appeared entitled 
to this distinction. According to these tables, the sun's 
mean longitude was 0°, or the mean vernal equinox oc- 
curred, in the year 1828, on the 22d of March at l h 2 m 59 s '05 
mean time at Greenwich, and therefore at l h 12 m 20 s -65 mean 
time at Paris, or l h 56 m 34 s -55 mean time at Berlin, at which 
instant, therefore, the equinoctial time was d h m S, 00, 
being the commencement of the 1828th year current of 
equinoctial time, if we choose to date from the mean 
tabular equinox, nearest to the vulgar era, or of the 6541st 
year of the Julian period, if we prefer that of the first year 
of that period. 

(938.) Equinoctial time then dates from the mean vernal 
equinox of Delambre's solar tables, and its unit is the mean 
tropical year of these tables (365 d -242264). Hence, having 
the fractional part of a day expressing the difference be- 
tween the mean local time at any place (suppose Green- 
wich) on any one day between two consecutive mean vernal 
equinoxes, that difference will be the same for every other 
day in the same interval. Thus, between the mean equi- 
noxes of 1828 and 1829, the difference between equinoctial 
and Greenwich time is d -956261 or d 22 h 57 m s -95, which 
expresses the equinoctial day, hour, minute, and second, 
corresponding to mean noon at Greenwich on March 23, 
1828, and for the noons of the 24th, 25th, etc., we have 
only to substitute Id, 2d, etc., for d , retaining the same 
decimals of a day, or the same hours, minutes, etc., up to 
and including March 22, 1829. Between Greenwich noon 
of the 22d and 23d of March, 1829, the 1828th equinoctial 
year terminates, and the 1829th commences. This happens 

9 On the instance of the author of these pages. 



OUTLINES OF ASTRONOMY 



845 



at d -236003, or at 6 h 51" 50 8 -66 Greenwich mean time, after 
which hour, and until the next noon, the Greenwich hour 
added to equinoctial time 364 d, 956261 will amount to more 
than 365*242264, a complete year, which has therefore to 
be subtracted to get the equinoctial date in the next year, 
corresponding to the Greenwich time. For example, at 
12 h m s Greenwich mean time, or d -500000, the equinoc- 
tial time will be 364-9562610+ -500000= 365-456261, which 
being greater than 365-242264, shows that the equinoctial 
year current has changed, and the latter number being 
subtracted, we get d -213977 for the equinoctial time of the 
1829th year current corresponding to March 22, 12 h Green- 
wich mean time. 

(939.) Having, therefore, the fractional part of a day for 
any one year expressing the equinoctial hour, etc., at the 
mean noon of any given place, that for succeeding years 
will be had by subtracting d, 242264, and its multiples, from 
such fractional part (increased if necessary by unity), and 
for preceding years by adding them. Thus, having found 
0-198525 for the fractional part for 1827, we find for the 
fractional parts for succeeding years up to 1853 as follows 10 : — ■ 



1828 


•956261 


1835 


•260413 


1842 


•564565 


1848 


•110981 


1829 


•713997 


1836 


•018149 


1843 


•322301 


1849 


•868717 


1830 


•471733 


1837 


•775885 


1844 


•080037 


1850 


•626453 


1831 


•229469 


1838 


•533621 


]845 


•837773 


1851 


•384189 


]832 


•987205 


■ 1839 


•291357 


1846 


•595509 


1852 


•141925 


1833 


•744941 


1840 


•049093 


1847 


•353245 


1853 


•899661 


1834 


•502677 


1841 


•806829 











10 These numbers differ from those in the Nautical Almanac, and would 
require to be substituted for them, to carry out the idea of equinoctial time as 
above laid down. In the years 1828-1833, the late eminent editor of that 
work, Dr. Young, used an equinox slightly differing from that of Delambre, 
which accounts for the difference in those years. In 1834, it would appear 
that a deviation both from the principle of the text and from the previous prac- 
tice of that ephemeris took place, in deriving the fraction for 1834 from that for 
1833, which has been ever since perpetuated. It consisted in rejecting the 
mean longitude of Delambre's tables, and adopting Bessel's correction of that 



846 



OUTLINES OF ASTRONOMY 



element. The effect of this alteration was to insert 3m 3s -68 of purely imagi- 
nary time between the end of the equinoctial year 1833 and the beginning of 
1834, or, in other words, to make the intervals between the noons of March 22 
and 23, 1834, 24h 3m 3s*68, when reckoned by equinoctial time. In 1835, and 
in all subsequent years, a further departure from the principle of the text took 
place by substituting Bessel's tropical year of 365-2422175 for Delambre's. 
Thus the whole subject fell into confusion. Under the present eminent super- 
intendent of the Nautical Almanac a compromise has been effected — a fixed 
equinoctial year of 365*242216 mean solar days has been adopted (and it is to 
be hoped will henceforward be adhered to), and corrections stated by which the 
data in the Almanacs for 1828-1834 may be brought into consistency with 
those in after years. According to this arrangement the fractional parts in 
question for 1828-1856 will run as follows: 



1828 


•958321 


1836 


•020593 


1843 


•325081 


1850 


•629569 


1829 


•716105 


1837 


.778377 


1844 


•082865 


1851 


•387353 


1830 


•473889 


1838 


•536161 


1845 


•840649 


1852 


•145137 


1831 


•231673 


1839 


•293945 


1846 


•598433 


1853 


■902921 


1832 


•989457 


1840 


•051729 


1847 


•356217 


1854 


•660705 


1833 


•747241 


1841 


.809513 


1848 


•114001 


1855 


•418489 


1834 
1835 


•505025 
•262809 


1842 


•567297 


1849 


•871785 


1856 


•176273 



[Note A on Art. 915. 

A rule proposed by Omar, a Persian astronomer of the court of Gelaleddin 
Melek Schah, in A.D. 1079 (or more than five centuries before the reformation 
of Gregory), deserves notice. It consists in interpolating a day, as in the Julian 
system, every fourth year, only postponing to the 33d year the intercalation 
which on that system would be made in the 32d. This is equivalent to omit- 
ting the Julian intercalation altogether in each 128th year (retaining all the 
others). To produce an accumulated error of a day on this system would re- 
quire a lapse of 5000 years, so that the Persian astronomer's rule is not only 
far more simple, but materially more exact than the Gregorian.] 

[Note B on Table, Art. 926. 

The civil epochs of the Metonic cycle and the Hejira are each one day later 
than the astronomical, the latter being the epochs of the absolute new moons, 
the former those of the earliest possible visibility of the lunar crescent in a 
tropical sky. M. Biot has shown that the solstice and new moon not only 
coincided on the day here set down as the commencement of the Callippic cycle, 
but that, by a happy coincidence, a bare possibility existed of seeing the cres- 
cent moon at Athens within that day, reckoned from midnight to midnight.] 



[Note C on Art. 932. 

The reformation of Gregory was, after all, incomplete. Instead of 10 days 
he ought to have omitted 12. The interval from Jan. 1, A.D. 1, to Jan. 1, 



OUTLINES OF ASTRONOMY 847 

A.D. 1582, reckoned as Julian years, is 577460 days, and as tropical, 577448, 
witli an error not excoediug 0d*01, the difference being 12 days, whose omis- 
sion would have completely restored the Julian epoch. But Gregory assumed 
for his fixed point of departure, not that epoch, but one later by 324 years, 
viz. Jan. 1, A.D. 325, the year of the Council of Nice; assuming which, the 
difference of the two reckonings is 9d*505, or, to the nearest whole number, 
10 days. To such as may have occasion, in an isolated case, to compute the 
interval between the beginnings of two proposed years, or the number of 
days elapsed from Jan. 1st Oh. of the one to Jan. 1st Oh. of the other, the 
following formulae will be found useful, in which it is to be observed that 

the notation = is used to express the integer portion only of the quotient 
n 

when one number m is divided by another n, and where D is the number of 
days required. 

Case 1. — Tears before Christ — from B.C. x to a.d. 1. 

D = a;.365 4- iL±J 
4 

Case 2. — Years after Christ — from a.d. 1 to a.d. x. 

x — 1 



(A) D=(a;_i). 365 4- 

(B) D=(»— 1). 3fi5 4- 2-Zzl — ^~ 20Q J-JL, 

v 4 100 400 

where the formula (A) is to be used if the later date (x) is before the change 
of style in that particular country for which the calculation is made, and 
(B) if after.] 

[Note J), Art. 223 a. 

(223 a.) Since the table of measured arcs in art. 216 was 
compiled, vast additions have been made to our knowledge 
of the true figure and dimensions of our globe; especially- 
through the extension of the two great arcs of Eussia and 
India, the former of which has been enlarged to an ampli- 
tude of 25° 20', the latter to 21° 21'. The whole series of 
geodesical measurements of any authority, executed in all 
parts of the world (these and the French arc, the next in 
magnitude, of course included), have been lately combined 
under one general and comprehensive system of calculation 



848 OUTLINES OF ASTRONOMY 

by Capt. A. E. Clarke, E.E., in an elaborate memoir (Mem. 
E. Ast. Soc. vol. xxix. 1860), of which the final result (in- 
cluding some slight subsequent corrections) may be stated 
as follows: 

The earth is not exactly an ellipsoid of revolution. The 
equator itself is slightly elliptic, the longer and shorter 
diameters being respectively 41,852,864 and 41,843,096 feet. 
The ellipticity of the equatorial circumference is therefore 
42 83 , and the excess of its longer over its shorter diameter 
about two miles. The vertices of the longer diameter are 
situated in longitudes 14° 23' E. and 194° 23' E. of Green- 
wich, and of its shorter in 104° 23' and 284° 23' E. The 
polar axis of the earth is 41,707,796 feet in length, and 
consequently the most elliptic meridian (that of long. 14° 
23' and 194° 23') has for its ellipticity ^ Vh) and the least so 
(that of long. 104° 23' and 284° 23') an ellipticity of S0 \. 3 . 

General Schubert also (Mem. Imp. Acad. Petersb. 1859) 
has arrived at a somewhat similar conclusion, by a totally 
different (and, as appears to us, less general and definitive) 
system of calculation. He makes the ellipticity of the 
equator 8 8 85 , and places the vertices of its longer axis 
26° 41' to the eastward of Capt. Clarke's. His polar 
axis, as deduced from each of the three great meridian 
arcs, the Eussian, Indian and French respectively, is 
41,711,019, 41,712,534 and 41,697,496 feet, the mean of 
which, giving to each a weight proportional to the length 
of the arc from which it is deduced, 11 is 41,708,710. 

A very remarkable consequence follows from these re- 
sults. If we reduce the polar axis last found to Brit- 

11 This is not Gen. Schubert's mode of procedure. He arbitrarily excludes 
the result of the French arc, and gives the Russian double the weight of the 
Indian — a procedure manifestly unfair. 



OUTLINES OF ASTRONOMY 849 

ish imperial inches, the result is 500,504,520, exceeding 
500,500,000 by 4520 in. On the other hand Capt. Clarke's 
result similarly reduced is 500,493,552, falling short of 
500,500,000 by 6448 in., so that we may take 500,500,000 
inches for the length of the earth's polar axis, with every 
probability of being within 150 yards of the truth. Were 
our imperial standard of length, then, increased by exactly 
one-thousandth part, the inch, foot, yard, etc., retaining 
their present relative proportions, our inch would then 
be, with all but mathematical precision, one five-hundred- 
millionth part of the earth's polar axis, a unit common to 
all the world. And, what is still more remarkable, if at 
the same time our standard of weight were increased by one 
2,600th part (or one-seventh of a grain on the ounce), an 
ounce of distilled water, at our present standard tempera- 
ture of 62° Fahr., would occupy precisely one-thousandth 
part of our (then) cubic foot, and our half -pint precisely 
one-hundredth. Thus, by a change such as would be ab- 
solutely unfelt in any commercial transaction, we should 
be put in possession of a modular or geometrical system 
(which we might decimalize if thought proper) far supe- 
rior both in principle and in accuracy to anything which 
has yet been devised on the important subject of a national 
system of weights and measures, the French metrical system 
not excepted.] 

[Note E on Art. 234. 

Mr. Broun of Edinburgh, and M. Babinet of Paris, have each separately and 
independently devised extremely ingenious applications of the principle of the 
torsion balance, to bring into equilibrium the force of gravity and the elastic 
force of a metallic spring, with a view to utilize the method here suggested of 
ascertaining the variation of gravity. It were much to be desired that both 
their methods should be brought to the test of practical trial.] 



850 OUTLINES OF ASTRONOMY 

[Note F, Art. 357. 
(357 a.) A series of concerted observations of the differ- 
ences of apparent declinations between the planet Mars and 
neighboring fixed stars, set on foot at the instance of M. 
Winnecke, during its last opposition, which took place 
under circumstances of proximity to the earth particularly 
favorable to the determination of its parallax, has resulted 
in assigning to that planet a parallax greater by about one- 
twenty-seventh part than that which would correspond to 
its distance as computed for the time by the hitherto as- 
sumed dimensions of the planetary orbits. The conclusion 
of course is, that these dimensions (including the earth's 
distance from the sun) have been overestimated by that frac- 
tional part of their value; that the sun's parallax, in place 
of 8* -6, or more exactly 8" -5766, should be set down at 
8 ,/ -8953, and its distance at 91,600,000 instead of 95,000,000 
miles. Now it is strongly corroborative of this conclusion- 
and, at the same time, affords a striking instance of the way 
in which the several departments of science depend on and 
illustrate one another, that a correction in the same direc- 
tion, and to nearly the same amount, is indicated by a re- 
cent redetermination, by direct experiment, of the velocity 
of light by M. Foucault, who finds that velocity not only 
less than that concluded by M. Fizeau's experiments (see 
art. 545), but even less than the commonly received esti- 
mate of 192,000 miles per second, by about the same frac- 
tional part. Light, as shown in art. 545, travels over the 
diameter of the earth's orbit in 16 111 26 s "6; and the time 
remaining thy same, a diminished speed corresponds to a 
diminished diameter, and therefore the sun's distance com- 
puted from the velocity so determined, and the time as 
given by observation, will come to be diminished in the 



OUTLINES OF ASTRONOMY 851 

same proportion. These and several other conspiring indi- 
cations lead to an extremely strong presumption that all the 
dimensions of our system have been overrated, and should 
be diminished by about one-twenty-eighth part. 

(357 b b.) In strong corroboration, or rather in full con- 
firmation, of this presumption, it must be especially noticed, 
that quite recently the whole subject of the reduction of the 
transit observations of 1769 has been resumed by Mr. Stone 
in a memoir for which the gold medal of the Astronomical 
Society for 1869 has been awarded to him, in which he has 
clearly shown that the received result of those observations 
has been vitiated by a misinterpretation of the expressions 
used by the observers in describing the phenomena of the 
external and internal contacts of the limb of Venus with 
that of the sun, which are complicated with certain optical 
appearances materially influencing the estimation of their 
times of happening; and that when those expressions are 
taken according to their real and legitimate meaning, and 
duly calculated on, they afford a value for the solar paral- 
lax of 8" -91 with a probable error of 0"*03 (in place of the 
hitherto received value, 8 //, 5776), agreeing very precisely 
with the value (8* *943) deduced by him from the assem- 
blage of comparative observations of Mars in his opposition 
of 1862, instituted at Greenwich, the Cape of Good Hope 
and Williamstown, Victoria, N. S. W". 

(357 c.) The distances being diminished in any ratio, 
the estimated masses will require to be diminished also, 
in the ratio of the cubes of the distances; for all the dis- 
tances will have to be reckoned on a new scale, and among 
the rest the diameters of the orbits of all satellites (the 
moon excepted); and as the squares of the periodic times 
are as the sums of the masses directly and the cubes of the 



852 OUTLINES OF ASTRONOMY 

distances inversely, the times remaining unchanged the 
masses must be diminished in the same proportion as those 
cubes. 

(357 d.) The time is not yet come for a complete and 
final determination of the exact "coefficient of diminution" 
to be applied to our planetary elements. The whole matter 
is too recent, and we must wait for the next transits of 
Yenus in 1874 and 1882 for a full and precise settlement 
of this important question. Meanwhile, therefore, and pro- 
visionally, the reader will bear in mind that in all our nu- 
merical statements of distances (those of the fixed stars and 
the velocity of light inclusive, but that of the moon ex- 
cepted), as well in the text as in the tables of elements, the 
numbers set down have to be diminished by one twenty- 
eighth of their value, and in the case of the masses (the 
moon's excepted) in the ratio of 27 3 : 28 3 or of 0*89664 to 1. 

(357 e.) The superficial reader (one of a class too numer- 
ous) may think it strange and discreditable to science to 
have erred by nearly four millions of miles in estimating 
the sun's distance. But such may be reminded that the 
error of 0"-32 in the sun's parallax, on which the correction 
turns, corresponds to the apparent breadth of a human hair 
at 125 feet, or of a sovereign at 8 miles off, and that, more- 
over, this error has been detected and the correction applied; 
and that the detection and correction have originated with 
the friends and not with the enemies of science.] 

{Note (x, Art. 387 a. 

This curious appearance of the "pores" of the sun's 
surface has lately received a most singular and unexpected 
interpretation from the remarkable discovery of Mr. J. 
Nasmyth, who, from a series of observations published in 



OUTLINES OF ASTRONOMY 853 

the Memoirs of the Lit. and Phil. Society of Manchester 
for 1S62, made with a reflecting telescope of his own con- 
struction under very high magnifying powers and under 
exceptional circumstances of tranquillity and definition, 
has come to the conclusion that these pores are the po- 
lygonal interstices between certain luminous objects of an 
exceedingly definite shape and general uniformity of size, 
whose form (at least as seen in projection, in the central 
portions of the disk) is that of the oblong leaves of a willow 
tree. These cover the whole disk of the sun (except in the 
space occupied by spots) in countless millions, and lie 
crossing each other in every imaginable direction. A rep- 
resentation copied from the figure in Mr. Nasmyth's memoir 
(the engraver being aided by photographs from his original 
drawings obligingly supplied, by him for the purpose) of 
the structure of the general luminous surface, is given in 
cur Fig. 1, Plate B, while Fig. 2 of the same plate exhibits 
their arrangement at the borders and in the penumbrae of 
s, spot. This most astonishing revelation has been con- 
firmed to a certain considerable extent, and with some 
modifications as to the form of the objects, their exact 
uniformity of size, and resemblance of figure, by Messrs. 
De la Rue, Pritchard, and Stone, in England, and M. Secchi 
in Rome. Mr. Stone compares them to rice grains, others 
to bits of straw. They strongly suggest the idea of solid 
bodies sustained in in equilibrio at a definite level (deter- 
mined by their density) in a transparent atmosphere passing 
by every gradation of density from that of a liquid to that of the 
rarest gas by reason of its heat and the enormous, superin- 
cumbent pressure (as in the experiments of M. Cagniard 
de la Tour on the vaporization of liquids under high pres- 
sure) ; their luminosity being a consequence of their solidity ; 



85i OUTLINES OF ASTRONOMY 

transparent and colorless fluids radiating no light from their 
interior however hot. 12 

(387 b.) In speaking of the intimate nature of the sun's 
luminous envelope, the remarkable phenomenon witnessed 
on the 1st of September, 1859, by two independent ob- 
servers, Mr. Carrington and Mr. Hodgson, ought not to 
be passed in silence. These two gentlemen viewing the 
sun, each at his own residence and without previous con- 
cert, at the same instant of time on that day, were both 
surprised by the sudden appearance, in the immediate 
confines of a large irregular spot, of what seemed to be 
two luminous clouds, much more brilliant than the general 
surface of the sun. They lasted about five minutes, and dis- 
appeared almost instantaneously, sweeping in that interval 
among the details of the spot, with' a visible progressive 
motion, over a space which could not be estimated at less 
than 35,000 miles. The magnetic needle, as was afterward 
ascertained, underwent a considerable and sudden disturb- 
ance at that very time; and the phenomenon, indeed, 
occurred during the continuance of one of the most re- 
markable and universal "magnetic storms" on record. 

(387 c c.) To Mr. Carrington, also, we owe a long- 
continued and most elaborate series of observations of the 
solar spots (contained in a memoir recently presented to 
the Eoyal Society), continued through a whole period oi 
their maximum and minimum frequency: arriving at the 
conclusion that the period of rotation of a spot is depen- 



12 Such is the view of their nature and of that of the solar photosphere sug- 
gested by the author in a paper "On the Solar Spots," published in the Quar- 
terly Journal of Science for April, 1864. M. Faye, in a Memoir read to the 
French Academy in January, 2865, has arrived at a conclusion nearly analo- 
gous. It ought to be noticed that Mr, Dawes still professes himself dissatisfied 
as to the existence of these objects. 



OUTLINES OF ASTRONOMY 855 

dent on its heliographical latitude; those on and near the 
sun's equator being carried round more swiftly than those 
in northern and southern latitudes. The empirical law at 
which he arrives as a mean expression of all his observa- 
tions assigns for the movement of rotation per diem of 
a spot in heliographical latitude I 

856—165' (sin I) 1 - 75 , 
so that a spot on the equator will make a complete sidereal 
revolution in 24 d '202—one in N. or S. latitude 15° in 25 d -44, 
and in 80° N. or S. in 26 d -24. 

(887 d d.) The confinement of the spots to a region lim- 
ited both ways in latitude and rarely beyond 30° on either 
side of the solar equator, as well as the frequency of their 
arrangement along parallels of latitude, has already (art. 
393) given us ground to conclude the existence of a circu- 
lation in the solar atmosphere relatively to the solid body 
of the sun, and to surmise an analogy between the cause of 
that circulation and that of the trade winds in our own 
atmosphere. It has been suggested 23 that, owing to obiate- 
ness in the solar atmosphere, its greater depth in the equa- 
torial regions might be conceived to oppose the free escape 
of heat more than in the polar, and so might give rise to 
a permanent inequality of temperature in the two regions 
from which movements analogous to those winds must of 
necessity originate. Were this the case, however, one of the 
results would (of equal necessity) be the production of an 
equatorial region of calm. Supposing our earth so covered 
with cloud that a spectator without could never see the 
solid body, and could judge only of its rotation from the 
observed circulation of dark masses of its clouds; he might, 

16 Results of Astronomical Observations at the Cape of Good Hope, p. 434. 



S56 OUTLINES OF ASTRONOMY 

it is true, conclude correctly the time of rotation of the 
solid body from observation of cloudy masses in the higher 
strata either on the equator or in high latitudes; but in the 
two intermediate zones north and south of the equator, he 
woald conclude a greater rapidity of rotation, owing to the 
general westerly tendency of the upper aerial currents in 
those zones. In proceeding then from the poles toward the 
equator, he would find, first, an apparent acceleration, then 
in certain latitudes N. and S. a maximum of rotary velocity, 
and thence up to the equator a comparative retardation. 
Mr. Carrington's law is therefore incompatible with this 
supposed analogy, and we must look elsewhere for its ex- 
planation. The only one which seems at all satisfactory is 
that of external force impressing such a movement; and 
thus we fall back upon the frictional impulse of circulating 
planetary matter in process of subsidence into and absorp- 
tion by the central body. The rotation of the sun, it will 
be remembered, is very much slower than that of a planet 
revolving just clear of its surface — a fact perfectly in unison 
with that theory of the formation of our system to which 
the term "the nebular hypothesis" (arts. 871, 872) has been 
applied, according to which the central body has resulted 
from the aggregation of all the matter assembling from 
every quarter, whose movements conflicting would destroy 
each other, leaving only as a surplus that small portion of 
rotation in one direction which remained outstanding. The 
same considerations render an equally plausible account 
of the intense heat of the central body.] 

[Note H, Art. 430 a. 

The enormous size and vast depth of many of the lunar 
craters, far surpassing, in both respects, anything observed 



OUTLINES OF ASTRONOMY 857 

on our globe, are certainly very striking features, but are 
easily reconcilable with what we know of the special condi- 
tions which obtain on the moon's surface. For while on 
the one hand the force of volcanic explosion and ejection 
is nowise dependent on the total mass of the planetary body 
en which the volcano may subsist, the repressing power to 
prevent an outbreak, which is the weight of the incumbent 
matter, is only about one-sixth of what an equal mass of 
overlying matter would exert on the earth, the force of 
gravity on the moon being less than that on the earth 
in that proportion. Again, when disrupted and scattered 
in fragments, the force generating their velocity of projec- 
tion being the same, and therefore also that velocity, the 
broken fragments, stones, scorise, etc., ejected would be 
hurled to far greater distances, being less powerfully coerced 
by gravity, in the same ratio. Most of the ejected matter 
would therefore fly out beyond the rims of the craters, in- 
stead of falling back to refill them. And lastly, we have 
to add, the absence of the resistance of an atmosphere— 
that powerful coercer of projectile range here on earth. 

(480 b b.) Changes in the aspect and configuration of 
particular portions of the lunar disk have been often sus- 
pected, but no satisfactory evidence had been obtained 
of anything which could not be accounted for, either by 
the difference of optical power in the telescopes used, or 
by difference of presentation to the solar rays, or to those 
reflected from the earth. Quite recently, however, on the 
16th of October, 1866, the crater marked as A in Lohr 
mann's Chart, (sec. iv.), and designated by Madler under 
the name "Linnseus" (a crater five miles and a half in 
diameter, and very deep, and which had served those 
selenographers as a zero-point of the first class for their 



858 OUTLINES OF ASTRONOMY 

micrometrical measurements), has been declared by Professor 
J. Schmidt, Director of the Observatory at Athens, to have 
altogether disappeared, and to be replaced by a smooth sur- 
face unmarked by any shadow! Subsequent observations 
made by him in November and December, under the most 
favorable circumstances as to solar illumination, failed to 
show any signs of the missing crater, though other much 
smaller ones in the neighborhood were readily perceived. 
The most plausible conjecture as to the cause of this dis- 
appearance seems to be the filling up of the crater from 
beneath by an effusion of viscous lava, which, overflowing 
the rim on all sides, may have so flowed down the outer 
slope as to efface its ruggedness and convert it into a grad- 
ual declivity casting no stray shadow.] 

[Note I on Art. 437. 

Our Plate C exhibits the appearance of a very rough and volcanic portion 
of the moon's surface as modelled from telescopic observation by Mr. Nasmyth, 
the engraving being taken from the photograph of the original model kindly 
furnished by him for the purpose. A very ingenious idea of Mr. Wheatstone 
has enabled the photographer to produce stereoscopic views of the moon, pre- 
senting it, not as a flat disk, but as a sphere, with all the mountains in full 
relief, and with all the appareance of a real object. Owing to the libration of 
the moon (art. 435) the same point of her surface is seen sometimes on one side 
of the centre of her disk, and sometimes on the other, the effect being the same 
as if, the moon remaining fixed, the eye were shifted from right to left through 
an angle equal to the total libratioD. Now this is the condition on which stere- 
oscopic vision depends: so that by choosing two epochs in different lunations in 
which the moon shall be presented in the two aspects best adapted for the pur- 
pose, and in the same phase of illumination (which the annual motion of the 
earth renders possible, by bringing the moon to 1he same elongation from the 
sun, in different parts of her own elliptic orbit), and taking separate and inde- 
pendent photographs of it in each aspect, the two stereoscopieally combined, so 
completely satisfy all the requisite conditions as to show the spherical form just 
as" a giant might see it whose stature were such that the interval between hia 
eyes should equal the distance between the place where the earth stood when 
one view was taken, and that to which it would have to be removed (our moon 
being fixed) to get the other. Nothing can surpass the impression of real cor- 



OUTLINES OF ASTRONOMY 859 

poreal form thus conveyed by some of these pictures as taken by Mr. De ia Rue 
with his powerful reflector, the production of which (as a step in some sort 
by man outside of the planet he inhabits) is one of the most remarkable 
and unexpected triumphs of scientific art. 

Mr. Birt has recently bestowed much pains on the frequent and minute 
scrutiny of particular and limited regions of the lunar surface.] 



[Note J. 

(859 a.) It is not necessary that the companion body pro- 
ducing these disturbances of proper motion should be non- 
luminous. Nothing prevents that it should be a small 
companion star which, though invisible to ordinary tele- 
scopes, or lost in the brightness of the disturbed luminary, 
r»ay become visible through telescopes of increased power. 
An tares has such a minute companion at only 12" distance, 
a Lyrse at 43*, and Procyon at 46". A disturbance in the 
regular progression of proper motion of the conspicuous 
star, periodical in its nature, both in E. A. and in N. P. D., 
would arise from the displacement of the larger star around 
the common centre of gravity, however comparatively mas- 
sive, and (mass for mass) greater the greater the distance of 
the individuals. It was not, therefore, without much inter- 
est that astronomers received the announcement of the re- 
cent detection of a small companion of Sirius, by Mr. Alvan 
Clarke, a most eminent and successful constructor of large 
achromatic telescopes, by means of an instrument of this 
description constructed by him of 18 inches aperture. Ac- 
cording to Messrs. Eutherford, Bond and Chacornac, this 
companion is at present situated at about 10'' distance, 
nearly following Sirius. It remains to be examined, how- 
ever, by a series of observations of distance and position 
continued for many years, 1st, Whether this star really is 
a satellite or binary companion of the principal star — a con- 



860 OUTLINES OF ASTRuNOMY 

elusion to which some have jumped (not unnaturally); and, 
2d, Whether, if so, its attraction will explain the observed 
inequalities of proper motion, which by no means follows 
of course: far more so, as M. Goldschmidt assures us that, 
with a telescope of very inferior power to that of Mr. 
Clarke, he has detected no less than six small companions 
of Sirius at distances from 10* to 60". Should this be veri- 
fied, we have our choice of disturbing influences; and it 
would be hard if the minute displacements, respecting 
which Messrs. Auwers and Peters agree (or any other) 
could not be plausibly explained antecedently to the indis- 
pensable verification of a real physical connection. Mean- 
while the conclusions of these geometers rest upon observed 
inequalities in E. A. only. Prof. Safford, of Ann Arbor, 
U. S., has, however, investigated those in polar distance, 
and finds that they are alike reconcilable with an elliptic, 
orbital motion, with one not incompatible with that previ- 
ously assigned, and with the hypothesis that the newly dis- 
covered star, so far as it has yet been observed, is really 
the disturbing body. 

(859 b b.) Spectrum analysis (see Note M) has of late 
been applied by various observers with great diligence and 
success to the light of the fixed stars, and even to that of 
the nebulae. Those of the stars are found to exhibit (vari- 
ously for different stars) many of those fixed lines which 
are considered characteristic of chemical elements such as 
are found in our planet. In applying this method of exam- 
ination to Sirius, however, Mr. Huggins has found that the 
brightest of three lines characteristic of hydrogen corre- 
sponds to a position in the spectrum of that star very 
slightly differing from the position of the same line in the 
solar spectrum, and that, in point of fact, the index of re- 



OUTLINES OF ASTRONOMY 861 

fraction of the prism employed for that line in the stellar 
spectrum is less, by a minute but measurable quantity, than 
in the solar. To interpret this observation, it must be re- 
membered that the undulatory theory of light postulates, 
as a condition indispensable to its interpretation of the 
different refrangibility of the rays, that the velocity of 
propagation of a luminous undulation within the refract- 
ing medium shall depend (according to some law depending 
on the physical nature of the medium) on its wave-length on 
arriving at the refracting surface; or, in other words, on the 
number of its undulations per second which are incident on 
that surface; the longer waves or less number per second 
corresponding to the lower degree of refrangibility. Sup- 
pose now a certain ray to originate in Sirius from some 
vibratory movement of a particle of its matter producing 
isochronous impulses of a certain determinate frequency 
on the luminiferous ether. Sirius being at any definite dis- 
tance from the earth at the moment of the first impulse, 
that impulse will reach the prism at a certain moment; and 
were the star and earth relatively at rest, its successors 
would follow it up and reach the refracting surface at in- 
tervals of time precisely equal to those of their origination. 
But if the star and earth be receding from each other with 
any uniform velocity, the next succeeding impulses, having 
each in succession a greater and greater distance to travel, 
will require a longer and longer time to arrive; or, in other 
words, the intervals of time between the arrivals of succes- 
sive impulses will exceed the intervals between their origi- 
nal production, each by as much time as it would take a 
ray of light to travel the distance run over by Sirius from 
the earth between two consecutive impulses. Arrived at 
the prism then, the ray consisting of these vibrations will 



862 OUTLINES OF ASTRONOMY 

have a lower refrangibility than if the distance of the star 
from the earth had remained unaltered ; and the amount of 
this difference being ascertained (by a most nice and deli- 
cate process of observation), the ratio of the relative ve- 
locity of recess of the star from the earth to that of light 
can be ascertained. Eeferring for details to Mr. Huggins' 
memoir, 14 we may state, as his final conclusion, 41*4 miles 
per second for this relative velocity, of which, at the time 
of observation, 12 miles per second were due to the mo- 
tion of the earth in its orbit; leaving 29 m, 4 per second, or 
2,540, 000 m daily, for the increase of distance between Sirius 
and oar system. The validity of this conclusion rests, of 
course, on the assumption (for in the absence of observation 
of other lines in the spectrum it can be only such) that the 
fixed line observed is due to hydrogen, and not to some 
other (unknown) chemical element.] 

[Note K, Art. 896 a. 

Several objects observed as nebulae are now missing 
from the heavens. They are such as have been (for the 
most part) only once observed, and may reasonably be sup- 
posed to have been telescopic comets. This, indeed, has 
in one instance proved to have been really the case; as, by 
tracing back the path of the 2d comet of 1792 to the date 
of the observation of a nebula discovered by Maskelyne on 
Feb. 14, 1793, but which is now missing, it appears to have 
really occupied that place (R. A. 2 h 39 m , N. P. D. 46° 15') 
at that time. But, besides these, there are cases in which 
a nebula, undoubtedly such, has either disappeared and 
reappeared in the same place, or has undergone some 

14 Phil. Trans. 1868. 



OUTLINES OF ASTRONOMY 863 

remarkable change of brightness; or, lastly, has been 
observed as a conspicuous object in a part of the heavens 
so well known as to make it exceedingly improbable that 
it should have escaped all previous observation. 

(896 b.) On the 11th Oct., 1852, Mr. Hind discovered a 
nebula in Taurus, previously unnoticed — in K. A. 4 h 14 m , 
1ST. P. D. 70° 49' (1860). lie saw it repeatedly, and in 1855 
and 1856 it was reobserved by M. D 1 Arrest. On the 3d 
Oct., 1861, M. D'Arrest missed it. "Hujus nebulas . . . 
ne umbram quidem," he says, "detegere valeo. Attamen 
semel ac saepius a me annis 1855 et 1856 observata est, 
ej usque locus quater determinatus." On Dec. 29, 1861, it 
was again seen, though with the utmost difficulty, in the 
great Pulkova refractor by M. Otto Struve, after which 
it had so far increased in brightness on March 22, 1862, 
as to bear a faint illumination of the wires. 

(896 c.) On Sept. 1, 1859, Mr. Tuttle discovered a nebula 
not previously observed, in E. A. 18 b 23 m 55 s , N. P. D. 
15° 29' 48" (1860). This nebula is described by M. Auwers 
as pretty bright and elongated in form. On the night of 
Sept. 24, 1862, it appeared to M. D'Arrest so brilliant and 
remarkable that he considers it impossible it should have 
been overlooked (if then so conspicuous) in the sweeps 
made by my father and myself over that part of the 
heavens. 

(896 d.) Mons. Chacornac has recently announced in the 
"Bulletin Meteor, de Paris," under date of April 28, 1863, 
the discovery of a nebula in Taurus, in K. A. 5 h 29 m 4 s , 
K P. D. 68° 52' 20" (1860), so conspicuous as to render 
its non-previous discovery most improbable, in a portion 
of the heavens so frequently under inspection, if always 
of its present brightness. 



864 OUTLINES OF ASTRONOMY 



e.) Certainly the last place in the heavens in which 
the discovery of a new nebula would have been expected, 
is within the cluster of the Pleiades. Yet here, close over 
Merope, one of the more conspicuous stars of that cluster, 
on Oct. 19, 1859, Mr. Tempel observed a large bright neb- 
ula which he took for a comet, and was only undeceived 
when, on observing it next night, he found it unchanged 
in place. On Dec. 31, 1860, it was seen, though with some 
difficulty, by himself and Dr. Pope, with the six-feet re- 
fractor at Marseilles. Its place for 1860 is R. A. 3 h 37 m 52 8 , 
N. P. D. m° 40' 13". M. Auwers describes it as 15' in ex- 
tent and triangular in form — but conceives that it might 
have escaped previous notice by reason of its proximity 
to so bright a star as Merope. Mr. Hind states also that 
he has often suspected nebulosity about some of the smaller 
outlying stars of the Pleiades. 

(896/.) Not less singular and startling is the observa* 
tion by Mr. Pogson of the bright and very conspicuous and 
well-known nebula, the 80th of Messier's Catalogue, often 
observed, and described as a compressed and beautiful 
globular cluster of very minute stars, in R. A. 16 h 8 m 41 s , 
N. P. D. 112° 37' 34" (1860). While examining the neigh- 
borhood of this object on May 28, 1860, his attention was 
arrested "by the startling appearance of a star, 7*8 m , in the 
place which the nebula had previously occupied." He had 
seen the nebula so recently as May 9, with the same tele- 
scope and power, and it presented nothing unusual. On 
June 10th the stellar appearance had vanished, but the 
cluster yet shone with unusual brilliancy and condensa- 
tion. Prof. Luther and M. Auwers had also perceived the 
change so early as May 21st, when it was rated as a star 
of the 6*7 magnitude. On June 10th, the nebula had dis- 



OUTLINES OF ASTRONOMY 865 

appeared to Mr. Pogson, though M. Auwers never quite 
lost sight of it, and could perceive that the star was excen- 
tric. The occurrence of a temporary or a variable star in 
so peculiar a situation is assuredly very remarkable. 

(896 g.) Lastly, by a letter from Mr. E. B. Powell, of 
Madras, an observer- of too much experience and. note 
to be easily deceived or to speak on light grounds, I am 
informed that the southern end of the very remarkable 
lemniscate-shaped vacuity close to the bright central star in 
the nebula about -q Argus (see Plate TV. Jig. 2), which, when 
the drawings were made from which that figure was taken, 
was closed and terminated by a strong and sharply cut out- 
line (marked by a small star in the upper edge of that va- 
cuity), is now decidedly open! More recent observations, 
however (by Lieut. J. Herschel), with an achromatic tele- 
scope of five inches aperture, accompanied with careful 
drawings of the appearance of the nebula (taken on Nov. 
22-3, 1868), make it evident — 1. That the lemniscate still 
exists, as such, though (as might be expected) not so 
strongly defined as when seen with an instrument of su- 
perior power; 2. That the relative situations of 48 out 
of 49 stars in the immediate vicinity of >?, laid down in 
the drawings, in respect of the principal star, have under- 
gone no material change, the 49fch being a very minute star 
of doubtful identity; and, lastly, that the principal star ^ 
itself, though greatly diminished in lustre, occupies most 
decidedly its old situation, pretty deeply immersed in the 
brightest portion of the nebula on the following or east- 
ern side of the lemniscate, and not (as stated in the last 
edition of this work, on what we considered sufficient 
authority) within the lemniscate or its remains, and out 
of the nebulosity. 



866 OUTLINES OF ASTRONOMY 

This is perhaps the right place to mention that a general 
catalogue of nebulae and clusters of stars (5,078 in number) 
in order of R. A., and brought up to 1860, with precession 
for 1880, and descriptions, prepared by the author of this 
work, has been published by the Royal Society as Part I. 
of the Philosophical Transactions for 1864. 

(896 h h.) Spectral observation, as already mentioned, 
has been recently applied to the brighter nebulae. The 
light, even of the brightest of these objects, is so exces- 
sively feeble (for it will be borne in mind that telescopes 
afford no means of increasing the intrinsic brightness of a 
surface) that any perception of delicate, hair-like, dark 
lines in their spectra like the fixed lines in that of the sun, 
is not to be expected. The phenomena which they exhibit, 
however, are very peculiar, and more in analogy with those 
of flame, or incandescent gases, than with solar or stellar 
sources of light. The brighter globular cluster, indeed, 
and those nebulae of irregular forms, which are either 
clearly resolved, or evidently of a resolvable character, 
give stellar spectra, i.e. trains of light of all gradations of 
refrangibility. It is otherwise with many of those nebulae 
of Sir W. Herschel's 4th class, to which the designation 
"Planetary Nebulae" has been applied, and with some 
others of an irresolvable character, among which are the 
great nebulae in Orion and Argo. The light of these is 
either monochromatic, that is, consisting of rays of one 
definite refrangibility (corresponding in all of them hitherto 
observed to the nitrogen line in the solar spectrum, or to 
the light emitted by nitrogen gas rendered incandescent 
by the electric discharge), or composed of this and of two, 
and in some few cases of three, other such monochromatic 
rays, one of which corresponds to one of the hydrogen 



OUTLINES OF ASTRONOMY 867 

lines of the solar spectrum. Such, in brief summary, are 
the remarkable and important results obtained by Mr. 
Huggins, 15 and fully corroborated by the observations of 
Lieut. J. Herschel, R.E., made at Bangalore, with the great 
advantage of an Indian sky, by the aid of a spectroscopic 
apparatus furnished (with the telescope already mentioned) 
by the Royal Society for observation of the solar eclipse 
of August 18, 1868. 16 If any hesitation should remain as to 
the certainty of conclusions from the scrutiny of objects 
so excessively faint, it will be removed by a fact recorded 
by the last-named observer, viz. — that on removing the slit 
or limiting aperture of the spectroscope, and viewing through 
the prism the whole field of the telescope directed to Mes- 
sier's 46th cluster, a rich and brilliant assemblage of stars, 
including among them the planetary nebula H. IV. 39, the 
latter was seen as a faint patch of light in the midst of an 
infinity of streaks, the continuous spectra of the individual 
stars. "Nothing," he remarks, "could have been more 
conclusive as a test." 17 Had the light of the nebula not 
been monochromatic or nearly so, its dilatation by the 
prism would have precluded its being seen at all as a 
definite object.] 

[Note L, Art. 858 a. 

A fundamentally different method of treating the prob- 
lem of the sun's proper motion from any of those described 
in art. 857, has been adopted by the present Astronomer 
Royal, by which the assumption of an approximate knowl- 



15 Phil. Trans. 1864, 1868. 

16 Proceedings of the Royal Society, xvi. p. 451, and xvii. pp. 58, 103. 
11 Ibid. xvii. 306. 

Astronomy — Vol. XX — 18 



OUTLINES OF ASTRONOMY 

edge of the situation of the "solar apex," is altogether 
dispensed with. It consists in referring the absolute proper 
annual motions both of the sun and the stars used in the 
inquiry, to linear co-ordinates fixed in space, and treating 
the question as a purely geometrical one, according to the 
rules of the calculus of probabilities, basing his procedure 
on two distinct assumptions, between which the truth must 
lie, viz. — 1. That all the "irregularities of proper motion" 
(meaning thereby all the residual amounts of annual move- 
ment in each case, which are not accountable for by solar 
motion) are mere results of error of observation and are 
not caused by any real motions in the stars. This is evi- 
dently an extreme supposition. — 2. That none of such re- 
sidual movements are due to error of observation, but 
all originate in real stellar movement. This is as clearly 
an extreme supposition the other way. Assuming then 
M. Struve's classification of the stars according to a scale 
of distances which has at least no prima facie improbability, 
and using for the purpose those 113 stars of a catalogue 
prepared with great care by Mr. Main, and published in 
the "Memoirs of the Astronomical Society," which indi- 
cate great proper motion, he arrives at the following con- 
clusions as to the situation of the apex, and the annual 
parallactic motion of the sun as seen from a star of the 
first magnitude, on each of these two suppositions. 

h . „ .,. ( Solar apex in E. A. 256° 54'; K.P.D. 50° 31'. 

1st Supposition | Paralla £ tic motion r : m 

OA Q ... j Solar apex R.A. 261° 29'; N.P.D. 65° 16'. 

2d Supposition | Pairallac P tic motioa r . 912i 

So far as the situation of the apex is concerned, both 
these results stand in what (considering the nature of the 
subject) may be called good accordance with those of 



OUTLINES OF ASTRONOMY 869 

our art. 854. The parallactic motion (or which comes to the 
same, the actual velocity of the solar motion) is, indeed, 
much greater than that of art. 858. But this evidently re- 
sults from the restriction of the inquiry to stars of great 
proper motion. 

(858 b.) These results were deduced by Mr. Airy in a 
memoir communicated to the Roy. Ast. Soc. in 1859. The 
subject has since been resumed (avowedly in extension of 
the same principle to a larger list of stars), by Mr. Dunkin, 
who, using the same geometrical formulae, but basing his 
results on the observed proper motions of 1167 stars, 819 
in the northern and 348 in the southern hemisphere, of 
all magnitudes, and all (sensible) amounts of proper motion, 
arrives at the following results on either of Mr. Airy's two 
extreme suppositions: 

icf Q„«rw«;« rt » j Solar apex in R. A. 261° 14'; N.P.D. 57° 5'. 
1st Supposition | Paralia £ tlc motlon ,. 3346> 

9A q „™««;«™ i Solar apex 263° 44'; N.P..D. 65° 0'. 

2d Supposition | Paralla £ tic motion ».4103. 

Agreeing remarkably (as to the second supposition, which 
is by far the more reasonable of the two) with the other, 
and, as regards both, exhibiting an almost perfect accord- 
ance in respect of parallactic motion with M. Struve's re- 
sults as given in art. 858. 

(858 c.) A very extraordinary circumstance remains to 
be noticed. From the general agreement of all the results 
of the investigations of so many astronomers and mathe- 
maticians, entering on trie inquiry in such various ways, 
and employing such a multitude of stars so variously com- 
bined, there cannot remain a shadow of doubt either as to 
the reality of the solar motion, or as to its direction in space 
toward a point very near to R. A. 259°, K. P. D. 56°. As 



870 OUTLINES OF ASTRONOMY 

to its velocity there is also every reason to believe that 
it is not extravagantly over- or under-estimated in the 
statement above given. But when we come to ascertain 
by calculation how large a portion of the whole proper 
motions of the stars; how much of that general residuum 
or caput mortuum alluded to in art. 856, as left outstanding 
after precession, aberration, and nutation, have exercised 
their solvent influences on it, remains yet unaccounted 
for; we shall find it includes by far the larger part of the 
total phenomenon of stellar proper motion. The sum of 
the squares of the total residua (in seconds of arc), uncor- 
rected for the proper motion of the sun, for example, in 
Mr. Dunkin's 1167 stars are, in E. A., 78-7583, and in 
K P. D. 63-2668. And when corrected for the effect of that 
motion (so concluded), they are represented in K. A. by 
75-5831, and in N. P. D. by 60-9084. Ko one need be sur- 
prised at this. If the sun move in space, why not also the 
stars? and if so it would be manifestly absurd to expect 
that any movement could be assigned to the sun by any 
system of- calculation which should account for more than 
a very small portion of the totality of the observed displace- 
ments. But what is indeed astonishing in the whole affair, 
is, that among all this chaotic heap of miscellaneous move- 
ment, among all this drift of cosmical atoms, of the laws 
of whose motions we know absolutely nothing, it should 
be possible to place the finger on one small portion of the 
sum total, to all appearance indistinguishably mixed up 
with the rest, and to declare with full assurance that this 
particular portion of the whole is due to the proper motion 
of our own system.] 



OUTLINES OF ASTRONOMY 871 

[Xote iM on Art. 400, note™. 

The reference of the dark lines in the solar spectrum to absorptive action in 
the sun's atmosphere has of late received a most unexpected confirmation, and 
it may now be considered as almost certain that they owe their origin to the 
presence in that atmosphere of the vapors of metals and metalloids identical 
with those which exist here on earth. These vapors, or many of them, have 
been shown by Kirchoff, Bunsen and Fizeau to possess the singular property, 
when present in an unhurried (or metallic) state in a flame, of destroying in the 
spectrum of that flame rays of precisely the refrangibilities of those which they 
themselves when burning emit in peculiar abundance. Though there is some- 
thing so enigmatical as almost to appear self-contradictory in the facts adduced 
— the conclusion, especially as applied to the most conspicuous of all the lines 
(one double one in the yellow, marked D by Fraunhofer, and which owes its 
origin to sodium) seems inevitable. The spectra of some of the stars seem to 
indicate the presence of chemical elements not identifiable with any terrestrial 
ones.] 

[Note N, Art. 905 a a. 

(905 a a.) Since the publication of the later editions of 
this work, meteoric phenomena have engaged the assiduous 
attention of many zealous and devoted observers, and have 
acquired an especial interest from the reappearance, in 1866, 
of the great November display mentioned in art. 900, under 
circumstances which render it an epoch in what may hence- 
forward be very properly termed Meteoric Astronomy. 
Before giving any account of these, however, it will be 
proper to mention that by the exertions of a Committee 
of the British Association, consisting of Mr. Brayley, 
Mr. Glaisher, Mr. Greg, and Mr. A. S. Herschel, acting in 
conjunction with numerous coadjutors scattered over the 
area of our island (and in correspondence with M. Heis, 
Prof. Haidinger, and other continental observers and mete- 
orologists), observing on the plan originally followed by 
Benzenburg and Brandes (much improved, however, and 
singularly facilitated by the use of a series of charts spe- 
cially constructed on the gnomonic projection for the purpose 



872 OUTLINES OF ASTRONOMY 

by the last-named member of the committee), a vast collec- 
tion of observations for the determination of the heights of 
appearance and disappearance, the velocities, and paths, 
of individual meteors, has been accumulated, and a con- 
siderable number of additional radiant points correspond- 
ing to dates of periodic recurrence other than those of 
August 10 and November 13, determined. As regards the 
heights of appearance and disappearance, and the velocities 
(understanding, of course, the relative velocities resulting 
from the simultaneous motions of the earth and the me- 
teor), the general result of these observations seems to be: 
1st, to assign a height intermediate between 20 and 130 
British statute miles above the earth's surface, for that to 
which the luminous or visible portion of the trajectory of 
non- detonating meteors is confined, with average heights of 
first appearance and final disappearance of 70 and 54 miles 
respectively, so far corroborating the evidence afforJed 
by auroral phenomena of the extension of our atmosphere 
much beyond the usually assigned limits of 45 miles; 2dly, 
a relative velocity intermediate between 17 and 80 miles 
per second, with a general average of about 34 miles, fully 
bearing out the earlier conclusions of Benzenburg and 
Brandes; and 3dly, a series of radiant points and annual 
epochs of which the following (exclusive of those of Aug. 
10 and Nov. 13) are the most remarkable: 



Jan. 2d . 


. R.A. 234° . 


. N. dec. 51° 


April 20 th . 


" 211° . 


35° 


Oct. 18th . 


90° . 


16° 


Dec. 12th . 


" 105° . 


30° 



(905 b b.) As regards the November display: — On trac- 
ing back the records of meteoric phenomena so far as they 
have been preserved by history or tradition, it has been as- 



OUTLINES OF ASTRONOMY 873 

oertained (chiefly by the laborious researches of Prof. New- 
ton of New Haven, U. S.) that no less than twelve 18 such 
displays, well characterized, have been noticed and recorded 
as occurring from the year A.D. 902 onward down to 1833, 
both inclusive, viz. in the years A.D. 902, 934, 1002, 1101, 
1202, 1366, 1533, 1602, 1698, 1799, 1832, 1833: all which are 
comprised within a chain of epochs breaking the interval 
between 902 and 1833 into periods of 32, 33, or 34 years 
each, corresponding to an average of 33*24 (33J) years, or of 
four such occurrences in 133 years. As to the calendar 
dates of the displays, the earliest, in A.D. 902, bears the 
date Oct. 13, O.S., and the others advance (with some 
considerable irregularities) in the calendar up to 1833, 
Nov. 13, N.S. Converting these dates into Julian clays 
current (arts. 929, 930), we find them to be respectively 
2,050,799 and 2,390,879, the difference of which, 340,068 
days, exceeds 931 tropical years (=340,040 d ) by 28 days; so 
that the dates advance in the calendar at an average rate of 
28 days in 931 years, or almost exactly 3 days in a century. 
The general impression resulting from the intervals of 33 
and 34 years between the great displays of 1799 and 1832, 
1833, that a similar one might be expected in 1866 or 1867, 
was by this converted almost into a certainty; and on the 
strength of this induction a grand meteoric exhibition on 
the night between Nov. 13 and 14, 1866, was announced 
as almost sure to take place, and all observers were fore- 
warned to be on the watch. The verification of this predic- 
tion will be fresh in the recollection of our readers, and the 



18 That of A.D. 931, Oct. 16, O.S., is here omitted. It seems to have been 
but a feeble exhibition, an irregular precursor of the more normal one of Oct. 
14, 934. 



874 OUTLINES OF ASTRONOMY 

spectacle presented by the heavens on that night, though 
falling short of what the glowing and no doubt exaggerated 
descriptions of the phenomenon of 1799 might have led 
some to expect, was such as can never be forgotten by those 
who witnessed it. Those who were not so fortunate will do 
well to be on the watch on the same anniversary in the cur- 
rent year 1867. 19 

(905 cc.) Attention being especially directed to the situ- 
ation of the radiant on this occasion, it was fixed (in refer- 
ence to the ecliptic) in long. 142° 35'; lat. 10° 27' N., at a 
point between the stars C and e Leonis, and somewhat above 
the star marked x in that constellation in Bode's Catalogue. 
Now the longitude of the earth at that time, as seen from 
the sun, was 51° 28', so that the radiant (in confirmation of 
a remark made by Prof. Encke on the occasion of the dis- 
play of 1833), if projected on the plane of the ecliptic, would 
be almost exactly in the direction of a tangent to the earth's 
orbit at the moment, or in "the apex of the earth's way." 
Hence it follows, that, regarding each meteor as a small 
planet, it must have been revolving (in a retrograde direc- 
tion, so as to meet the earth) either in a circle concentric with 
the earth's orbit (a thing in itself most improbable, and 
which would bring about a rencounter every year, contrary 
to observation) or in an ellipse having either its perihelion 
or its aphelion coincident, or very nearly so, with the point 
of rencounter at the descending node, in longitude 51° 28'; 



19 On that occasion (Nov. 13 and 14, 1867) the principal display took place 
in longitudes much westward of our island. At Bloomington, Indiana, UVS., 525 
were seen by Prof. Kirkwood between midnight and 5h. 15m. A.M. Off Mar- 
tinique, they appeared as a brilliant shower, and at Trinidad, according to Com- 
mander Chimmo, 1600 were counted between 2h. A.M. and daylight; while at 
Nassau in the Bahamas, Captain Stuart and his co-observers registered 1040 
between lh. 0m. and 5h. 34m. A.M. 



OUTLINES OF ASTRONOMY 875 

and as a necessary consequence with its major axis lying in 
or vorv nearly in the plane of the ecliptic. 

(905 d d.) Admitting the meteors to be revolving plane- 
tules, the recurrence of these rencounters at average intervals 
of 4 in 183 years is explicable on two distinct hypotheses as 
to the kind of ellipse described by the meteoric group. It 
may be either one very nearly approaching to a circle de- 
scribed in a period not very different from a sidereal year, 
or a very elongated one described in the exact period of 33J 
of such years. We will consider the cases separately. The 
first supposition admits of the adoption of two distinct ellip- 
ses: — 1st, that suggested by Prof. Newton, in which the ren- 
counter takes place at the aphelion of an ellipse described in 
354 d -57, or 10 d, 67 short of a sidereal year, corresponding to a 
semiaxis 0-981, and an excentricity 0*0204; or, 2dly, that 
proposed by the writer of an article on Meteoric Showers 
in the "Edinburgh Eeview" for January, 1867, where it is 
supposed to happen at the perihelion of an ellipse described 
in 376 d, 56, or ll d -33 more than a year — corresponding to a 
semiaxis 1*021, and an excentricity 0-0192. In the first of 
these ellipses, a meteor revolving would in each sidereal 
year gain 10° 50' in its orbit on a complete revolution, and 
in the other would lose as much; so that at the end of 33 
years, in the former case it would be found to have overshot 
the original point of rencounter by 2° 30', and in the latter to 
fall short of it by just so much: and tracing it round from 
revolution to revolution, it will be found in either case that 
after a series of intervals succeeding each other in the cycle 
33, 33, 33, 34 years, the meteor will always be found so near 
the original point of rencounter that an extension of the 
whole group so as to occupy 11° in their common orbit will 
render extremely probable, and one of 22° will insure its 



876 OUTLINES OF ASTRONOMY 

penetration by the earth at some point or other, with a 
probability of such penetration taking place twice in two 
successive years. 

(905 e e.) The other hypothesis, suggested by Sig. Schia- 
parelli (Director of the Observatory at Milan), is that of a 
ren counter at or very near to the perihelion of an ellipse of 
33i years, corresponding to a semiaxis 10*340, and an excen- 
tricity 0-9033; the rencounter in this, as in the other, taking 
place also at the descending node. Such a period, com- 
bined with an extent of the group on the orbit such as would 
occupy somewhat more than a year in passing through the 
node (i.e. ^ of its whole circumference), would bring round 
rencounters in precisely the same cycle of years, with a fair 
probability, and if of twice that extent with, a certainty of 
their happening. An increased extent of the group to 
somewhat beyond this would give rise to a frequent occur- 
rence of two or even three rencounters in annual succession, 
and would therefore cover the whole series of recorded 
instances. 

(905//.) The regular advance of three days per century 
in the calendar date of the phenomenon is partly accounted 
for by the greater length of the sidereal year (which brings 
the earth round to the same point in its orbit) as compared 
with the tropical year (which brings it to the same longitude, 
reckoned from the receding equinox, by which the calendar 
is regulated). This accounts for l d -4 per century; the re- 
maining l d, 6 must arise from a slow and regular advance in 
the place of the node to the amount of 1° 36' per century, 
or 57" *6 per annum, due probably to planetary perturbation, 
and chiefly, no doubt, to the disturbing action of the earth 
itself in its successive passages through the group. 

(905 g g.) On either of the two former orbits the velocity 



OUTLINES OF ASTRONOMY 877 

of the meteors will be very nearly equal to that of the earth 
in a retrograde direction, whence it will readily appear that 
the true inclination of the orbit will be almost exactly dou- 
ble the apparent — that is to say, 20° 54'. In the case of the 
long ellipse, the velocity of the meteor in perihelio will be 
found to be to that of the earth as 1-371 : 1, so that. suppos- 
ing AC to be the earth's orbit, and bd that of the meteor, the 




apparent inclination bad being 10° 27', and the sides BD, DA, 
respectively, 1-371 and 1, we shall find the angle dba=7° 13', 
and therefore the true inclination bdc = 18° 31'. 

(905 h h.) The supposition of minute planetary bodies re- 
volving in a nearly circular orbit of almost exactly the 
dimensions of the earth's in a retrograde direction, and at 
an inclination not greater than that of some of the asteroids, 
stands in such strong opposition to all the analogies of our 
system, as to render it in itself highly improbable; add to 
which, that (as no perturbative action could possibly have 
flung them from without into such an orbit) they must be 
supposed to have so circulated for countless ages, during 
which time their innumerable rencounters with the earth 
must have torn the group to pieces, and scattered all its 
members which escaped extinction into orbits of every dif- 
ferent inclination and excentricity. On the other hand, the 
ellipse of 33 y -J has a decidedly cometary character; and in 
such, retrograde motion is not uncommon. By a most 
singular coincidence, remarked almost simultaneously by 
Messrs. Peters and Schiaparelli (a coincidence too close 



878 



OUTLINES OF ASTRONOMY 



and striking to admit of hesitation as to their community 
of origin), the elements of the first comet of 1866, discov- 
ered by M. Tempel, coincide almost precisely, in every par- 
ticular except in the date of the perihelion passage, with 
those we have just derived from the very simple considera- 
tions adduced. The parallel is as below: 





Meteoric Orbit 






Tempel's Comet 


Perihelion passage . . 


Nov. 13, 1866 . . . Jan. 11, 1866 


Perihelion distance 


0-9893 20 






0-9765 


Excentricity 


0-9033 






0-9054 


Semiaxis major . . 


10-340 






10-324 


Inclination . „ 


18° 31' 






17° 18'-1 


Long, descending node . 


51° 28' 






51° 26'-l 


Periodic time . . 


33y-25 






33-176 


Motion 


Retrograde 






Retrograde 21 



(905 i i.) It will not fail to have been observed that a 
major semiaxis 10*34 with a perihelion distance 1 will throw 
the aphelion of the meteoric orbit to a distance from the 
sun=19*68, that is to say, but a short distance beyond the 
orbit of Uranus; while the fact of the axis major itself lying 
exactly or at least very nearly in the plane of the ecliptic, a 
plane itself very little inclined to the orbit of Uranus, will 
insure a very near appulse of the meteors to that planet 
whenever their two mean motions may have brought or may 
hereafter bring them to the corresponding parts of their 
orbits, allowing for the change (if any) in the position of 
the axis. We say, if any, for it is not of necessity the same 



20 This is the earth's radius vector on Nov. 13. 

21 The computations of Sig. Schiaparelli, founded on a somewhat different 
(and, we are inclined to think, less accurate) situation of the radiant of last 
November, lead him to assign the date Nov. 10 for the perihelion passage, and 
to the perihelion itself the longitude 56° 25' -9, again agreeing well with that of 
the comet in question, which is 60° 28'. But we have preferred (avoiding all 
niceties, which, in the actual uncertainty as to the exact place of the radiant, 
are, after all, premature), for the sake of perspicuity, to present the chain of 
reasoning m a form requiring almost no calculation. 



OUTLINES OF ASTRONOMY 879 

as that of the node, and, though calculable, has not as yet 
been calculated. This, however, does not affect the conclu- 
sion that such near appulse must at some former time have 
taken place, and will do so again. M. Leverrier, to whom 
these considerations seem to have occurred independently, 
has concluded that it did take place about the year A.D. 
126; and the motion of both bodies being very slow at that 
time (the velocity of the meteors being in aphelio only 5 07 
of that of the earth, or only 1-32 mile per second), they 
would remain for a long time within the influence of the 
planet's disturbing power, while at the same time that power 
would be acting at the greatest advantage to produce deflec 
tion from their line of motion. Hence that illustrious as- 
tronomer was led to conclude, that, just as Jupiter on a 
similar occasion seized on and threw into an orbit of short 
period Lexell's comet (see art. 585), so at that epoch a wan- 
dering group of planetules, whose existence would, but for 
that meeting, have never become known to us, was deflected 
into the ellipse they actually describe. Sig. Schiaparelli, 
on the other hand, considering that the semi-minor axis of 
the meteoric ellipse is but small (0*441), so that by reason 
of the moderate inclination the meteor in its course can 
never rise much more than l£ radius of the earth's orbit 
above its plane; appears disposed to attribute their present 
form of orbit to the attraction of Jupiter or Saturn, within 
whose disturbing influence he considers that they must at 
some period or other have passed, an opinion which appears 
to us less probable, inasmuch as the disturbing force would 
in that case have tended chiefly in a direction at right angles 
to the plane of their motion, and (by reason of the much 
greater velocities of both bodies) have acted for a much 
shorter time. 



880 



OUTLINES OF ASTRONOMY 



(905 jj.) For the meteors of the 10th of August, adopt- 
ing as the place of the radiant the star h Persei, assigned by 
the observations of Mr. A. S. Herschel in 1863, and assum- 
ing the orbit to be a parabola (an assumption which deter- 
mines the velocity at the moment of rencounter, being to that 
of the earth regarded as describing a circle, in the constant 
ratio of \/2 : 1), a velocity which he found to agree toler- 
ably well with that directly determined by Mr. Herschel 
and his co-observers on the same occasion, M. Schiaparelli 
has also computed the elements of their orbit. And again, 
by a coincidence hardly less striking, these are found to 
agree with the elements of the great comet of 1862, as the 
following comparison will show: 





August Meteors, 

SchiaparelLTs 

Elements. 


Comet lllo 1862. 

Elements of 

Oppolzer. 


Passage through descending Node 

Perihelion Passage 

Longitude of Perihelion . . . 
Longitude of Ascending Node 

Inclination 

Perihelion Distance 

Period 

Motion 


1866, Aug. 10-75 

— Julv 23-62 

343° 38' 

138° 16' 

64° 3' 

0-9643 

Retrograde 


1862, Aug. 22-9 

344° 41' 

137° 27' 

66° 25' 

0-9626 

123-74 

Retrograde 



Without supposing the orbit absolutely parabolic, an ellipse 
of long period (say 124 years) would equally well satisfy the 
conditions; but to make the rencounter annual, a complete 
annular or elliptic stream of meteors would be required. 
The radiant point of the August meteors, however, seems 
hardly so definite as that of the November group, the deter- 
mination in different years by different observers differing 
considerably. Both these considerations would seem to 
authorize the ascription of a far higher antiquity to the 
introduction of this assemblage into our system, giving 



OUTLINES OF ASTRONOMY 881 

time not only for the individual meteors to gain or lose 
upon each other in consequence of minute differences in 
their periodic time, so as to draw out the original group 
into a stream, but to disperse themselves over considerable 
differences of inclination and excentricitj by the effect of the 
earth's perturbative action, while all the phenomena of the 
November group point to a much more recent origin. 

[Note O, Art. 395. 

(395 b b.) The total solar eclipse of August 18, 1868, 
which, commencing not far from Aden, at the entrance of 
the Red Sea, traversed the whole peninsula of India from 
Malwa to Masulipatam, and pursued its course eastward and 
southward across the Malayan peninsula, to the extreme 
northern point of Australia, afforded an excellent oppor- 
tunity for the critical examination of the marginal protu- 
berances as well as the phenomena of the corona; which, 
if seen at all from a station on the central line, could not 
be held to originate in the earth's atmosphere by reason of 
the great breadth of the total shadow (at least 115 miles). 
Accordingly, it was eagerly seized, and competent ob- 
servers, well furnished with every requisite instrument 
and means of observation and record, took up their sta- 
tions at points on or very nearly adjacent to the central 
line. The unusual duration of the total obscuration, being 
nearly six minutes, allowed ample time for making all the 
necessary observations, as well as for securing photographs, 
which last desideratum was successfully accomplished at 
Guntoor in India, by skilled photographers under the direc- 
tion of Major Tennant, as also at Aden. The final results 
may be thus briefly stated. — 1. The darkness was by no 
means so great as was expected: doubtless, owing to the 



882 



OUTLINES OF ASTRONOMY 



great amount of light emitted by the corona and the mar- 
ginal prominences. The light of the former gave a con- 
tinuous spectrum, and moreover was found to be distinctly 
and strongly polarized; everywhere in a plane passing 
through the point examined and the sun's centre. This 
establishes beyond all possible doubt its origin in reflec- 
tion from a solar atmosphere exterior to the photosphere 
(or at least of some sort of envelope, whether atmospheric 
or nebulous in its nature and connection with the sun) of 
vast extent, though probably of small density, and is alto- 
gether opposed to its origination in that of our earth; our 
sky light so near the sun exhibiting no trace of polariza- 
tion. — 2. The red prominences, which were numerous and 
most remarkable, showed no sign of polarization, and were, 
therefore, self-luminous. One of them, the most conspicu- 
ous, projected like a horn or tall excres- 
cence to the distance of 3' 10" from the 
true limb of the sun, which corresponds 
to a vertical elevation of 90,200 miles 
above the level of the photosphere. Its 
outline, as photographed at Gruntoor, was 
as in the annexed diagram, indicating 
by its markings a spiral form, like that 
which might be conceived to result 
from a combined rotatory and ascen- 
sional movement, as of a vast column of 
ignited vapor rushing upward with a swirl 
from the photosphere into the higher regions of a non- 
luminous atmosphere. — 3. The light of the prominences 
subjected to spectroscopic examination was in accordance 
with this idea. It gave no continuous spectrum, but ap- 
peared to consist of distinct monochromatic rays, or definite 




OUTLINES OF ASTRONOMY 883 

Wight lines (characteristic of incandescent gases). Of these, 
M. Janssen, stationed at Guntoor, saw six, in the red, yel- 
low, green, blue, and violet regions of the spectrum; two of 
them corresponding to Fraunhofer's lines c, F, indicative 
of hydrogen. Major Tennant, at the same station, perceived, 
only four, viz. : c in the red, and D in the orange (corre- 
sponding respectively to hydrogen and sodium) — one in 
the green near F (hydrogen), and a fourth seen with diffi- 
culty in the blue near to G. Lieutenant Herschel at Jam- 
kandi (where the total phase of the eclipse was much inter- 
fered with by passing clouds) perceived distinctly three 
vivid lines, red, orange, and blue, and no others, nor any 
trace of a continuous spectrum. The orange line proved 
by measurement to coincide precisely with D, the others 
approximated to c and F, and probably, the difficult circum- 
stances of the measurements considered, were coincident 
with those lines. M. Eayet at Wah Tonne, in the Malayan 
peninsula, noted no less than nine brilliant lines correspond- 
ing to the solar dark lines B, D, E, b, F, two adjacent to G, 
and one between b and f (probably Barium). These obser- 
vations are quite decisive as to the gaseous nature and 
vehement incandescence of the prominences, and indicate 
astonishingly powerful ascensional movements of what 
might be called flame (were combustion possible) in an 
atmosphere reposing on the photosphere. — 4. Besides these 
hard and sharply denned prominences, were also seen 
ranges irregular in form, of what might perhaps be con- 
sidered cloudy or vaporous matter, of less intensity and 
softer outline. 

(395 c c.) Reasoning on the monochromatic character of 
the light emitted by incandescent gases, and speculating 
on the extreme probability of the solar prominences being 



884 OUTLINES OF ASTRONOMY 

in the nature of tumultuous ejections of such gases, it had 
early occurred both to Mr. Huggins and Mr. Lockyer that 
they might possibly become the subject of spectroscopic 
study, as appendages to the sun's limb, or in the umbrae 
of spots unilluminated by photospheric light, without the 
necessity of waiting for the rare occurrence of a total 
eclipse. Accordingly, during the two years immediately 
preceding that of August 18, 1868, the former made several 
attempts with various spectroscopic and other contrivances 
(such as viewing the projected image of the sun's border 
through combinations of colored glasses, etc.) to obtain 
a view of them, though without success; and the latter 
had applied for and obtained from the Eoyal Society a 
grant for the construction of an apparatus for the purpose, 
which, however, was not completed till after the occurrence 
of the eclipse. Meanwhile the actual observation of the 
monochromatic character of their light, and the exact co- 
incidence of their lines with situations which in the spec- 
trum of the photosphere are marked by a deficiency of 
light, at once suggested to M. Janssen, as it did also 
to Lieutenant Herschel, the possibility of discerning, or at 
all events of, as it were, feeling them out, the former 
by means of the spectroscope, the latter by combinations 
of colored glasses. M. Janssen, on the day immediately 
following the eclipse, put his conception into practice, and 
at once succeeded. Placing the slit of his spectroscope so 
as partly to be illuminated by the edge of the photosphere, 
and partly by the light (from whatever origin) exterior to 
it, he found the spectrum of the former portion in the imme- 
diate neighborhood of the ray C to be crossed (as might be 
expected) by that dark line. At the point of the limb to 
which his examination was first directed, nothing was seen 



OUTLINES OF ASTRONOMY 885 

beyond. But, on shifting the point of examination grad- 
ually along the limb, a small dot of ruddy light was per- 
ceived, in exact continuation outward of the dark line, 
which, on continuing the movement of the spectroscope 
along the limb in the same direction, gradually lengthened, 
and then again shortened: thus revealing the existence of 
a prominence giving out that particular monochromatic red, 
whose form and outline he was thus enabled to trace out. 
Directing his attention, in like manner, to the dark line F, 
the same phenomenon was repeated, in the tint proper to 
that region of the spectrum. At some points it was also 
observed that the bright line of the prominence encroached 
upon and extended into the corresponding dark line of the 
photosphere. 

(395 d d.) Mr. Lockyer's apparatus having meanwhile 
been completed, he was at length enabled to announce 
(on October 20) that after a number of failures which made 
the attempt seem hopeless, he had at length succeeded in 
observing, as part of the spectrum of a solar prominence, 
three bright lines, one absolutely coincident with C, one 
near D, and one nearly coincident with F. On February 
16, 1869, another practical step in the same direction was 
made by Mr. Huggins, who, limiting, by an ingenious 
contrivance, the light admitted to his spectroscope to rays 
of about the refrangibility c, widening the slit sufficiently 
to admit of the whole prominence being included in its 
field, and absorbing the light of other refrangibilities so 
admitted by a ruby glass, was enabled distinctly to per- 
ceive at one view, the form of the prominence. Almost 
immediately after, Mr. Lockyer succeeded, by merely widen- 
ing the slit of his spectroscope, without the use of any absorjJ- 
tive media, in obtaining a clear view of the forms in ques- 



886 OUTLINES OF ASTRONOMY 

tion. "The solar and atmospheric spectra being hidden, 
and the image of the wide slit alone being visible, the 
telescope or slit is moved slowly, and the strange shadow- 
forms flit past. Here one is reminded, by the fleecy infi- 
nitely delicate cloud-films, of an English hedgerow with 
luxuriant elms; here of a densely intertwined tropical 
forest, the intimately interwoven branches threading in all 
directions; the prominences generally expanding as they 
mount upward, and changing slowly, indeed almost imper- 
ceptibly." Lastly, on the 4th, 5th and 6th of May, Lieu- 
tenant Herschel found the spectrum of the solar envelope 
to be visible without difficulty, and without other aid than 
the spectroscope adapted to his telescope, and was enabled 
to form a general picture of the distribution of the luminous 
region surrounding the sun. Two prominences were in par- 
ticular examined, one of which formed a luminous cloud 
floating 1' or 2' above the surface. He perceived also (now 
for the first time) a fourth line near G- (since seen re- 
peatedly), and subsequently another between F and G. On 
this last occasion, having at first swept round the sun and 
found nothing particularly worthy of remark, on returning 
to the point of departure, he perceived the lines much more 
brilliant and intense than usual, and further scrutiny satis- 
fied him that he had been witness to a "violent and spas- 
modic eruption of vapor" lasting only a few minutes! 
The mode in which he was enabled thus to discern the 
forms of the solar clouds consisted in giving to the tel- 
escope a vibratory motion up and down, on the principle 
of the persistence of luminous impressions on the retina, 
by which the perception of the total form of an object re- 
sults from the mental combination of a series of linear sec- 
tions of its area. And he describes the appearance of these 



OUTLINES OF ASTRONOMY 887 

solar clouds as "very similar to terrestrial — fleecy, irregular- 
shaped, and illuminated; just such as eclipses have told us 
they are." We have thus a new chapter of solar physics 
opened out, the commencement, doubtless, of a series of 
grand discoveries as to the nature and constitution of the 
great central body of our system. Mr. Huggins has also 
applied spectrum analysis to the comae and tails of comets 
in which he considers satisfactory proof to exist of the 
presence of carbon. 



LIST OF PLATES 



Plate I. Fig. 1. Faculse of the Sun. 
Fig. 2. Spots on ditto. 

Fig. 3. Appearance of ditto in a total Eclipaa. 
Figs. 4, 5. Spots as seen by*Mr. Dav.es. 

Plate II. Fig. 1. Messier 's 13th Nebula resolved into Stars. 
Fig. 2. The Comet of 1818. 
Fig. 3. The Nebula in Andromeda. 

Plate III. Fig. 1. Mars as seen August 16th, 1830. 

Fig. 2. Jupiter as seen September 23d, 1832. 

Fig. 3. Saturn showing the interior rings and belts. 

Plate IV. Fig. 1. The great Nebula in Orion. 

Fig. 2. The great Nebula in Argo. 
Plate V. Fig. 1. Nebula (30 Doradus) in the Nubecula Major. 

Fig. 2. Lunar Volcano, as shown by a 20 -feet reflecting Telescope, 
aperture 18 inches. 

Plate VI. Fig. 1. Various Appearances of Halley's Comet at its last 

Apparition. 
Fig. 2. Double Comet of Biela, as seen on its last return. 
Fig. 3. Messier's 51st Nebula as shown by Lord Posse's great 

Reflector. 

Plate A. Figures illustrative of the Perturbations of Uranus by Neptune. 

Plate B. Fig. 1. The willow-leaved structure of the sun's photosphere. 
Fig. 2. The same in the neighborhood of a spot. 

Plate C. A portion of the Moon's surface, from a model by Mr. Nasmyth. 



(888) 



APPENDIX 



891 



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892 



APPENDIX 



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APPENDIX 



II. SYNOPTIC TABLE OF THE ELEMENTS OF THE PLANE- 
TARY SYSTEM 

N.B. — « denotes the mean distance from the sun, that of the earth being 
taken for unitv ; P the mean sidereal period in mean solar days; e the excen- 
tricity in decimal parts of the semiaxis; i the inclination of the orbit to the 
ecliptic; Q the longitude of the ascending node; w that of the perihelion from 
node on orbit; L the mean longitude of the planet at the moment of the epoch 
E, for which the elements are stated; M the denominator of the fraction ex- 
pressing the mass of the planet, that of the sun being 1 ; D the diameter in 
miles ; A the density, that of the earth being 1 ; T the time of rotation on its 
axis; d the mean angular equatorial diameter of the body of the planet, at its 
mean distance from the earth, in seconds; « the elliptieity of the spheroid, as a 
fraction of the equatorial diameter; y the inclination of the axis of rotation to 
the plane of the ecliptic; H the mean intensity of light and heat received from 
the sun, that received by the earth being 1. The asteroids are numbered in 
their order of discovery. 







a 


P 


e 


i 










O 1 It 


o 


The Sun 










s 


Mercury . 


0-3870981 


87-9692580 


0-2054925 


f 9-1 


6 


Venus . 






0-7233316 


224-7007869 


0-0068722 


3 23 28-5 


© 


Earth . 






1-0000000 


365-2563612 


0-0167918 




S 


Mars . 




i i 


686-9796458 


0-0931125 


1 51 6-2 


l 


Ceres . 






2-7664313 


1680-650 


0-0805096 


10 36 27-8 


2 


Pallas . 






2-7686906 


1683-103 


0-2400116 


34 43 8-7 


3 


Juno 






2-6707543 


1594-221 


0-2565949 


13 1 9-0 


4 


Yes la . . 






2-3616682 


1325-640 


0-0890879 


7 7 55-7 


5 


Astreea . 






2-5778960 


1511-800 


0-1872123 


5 19 5-7 


6 


Hebe . , 






2-4246290 


1379-007 


0-2028856 


14 46 38-0 


7 


Iris . . . 






2-3860360 


1346-212 


0-2312145 


5 27 53-3 


8 


Flora . 






2-2013860 


1193-004 


0-1567040 


5 53 8-0 


9 


Metis . . 






2-3860333 


1346-211 


0-1238156 


5 36 7-3 


10 


Hygeia . 






3-1511972 


2043-200 


0-0993911 


3 48 47-1 


11 


Parthenopc 


. 


i 2-4524062 


1402-770 


0-0990197 


4 37 0-9 


12 


Yictotia 






2-3344918 


1302-756 


0-2178214 


8 23 1-5 


13 


Egeria . 






2-5762970 


1510-400 


0-0867520 


16 31 54-9 


14 


Irene . . 






2-5863965 


1519-288 


0-1686725 


9 7 27-3 


15 


Eunomia 






2-6441373 


1570-452 


0-1868308 


11 43 35-2 


16 


Psyche . 






2-9263840 


1823-493 


0-1341254 


3 3 58-6 


17 


Thetis . 






2-4732811 


1420-721 


0-1276441 


5 36 5-6 


18 


Melpomene 






2-2956321 


1270-430 


0-2174420 


10 8 58-6 


19 


Fortuna 






2-4415662 


1393-481 


0-1572326 


1 32 30-2 


20 


Massilia 






2-4089122 


1365-621 


0-1442977 


41 ll'l 


21 


Lutetia 






2-4347444 


1387-643 


0-1617091 


3 5 21-0 


22 


Calliope 






2.9094987 


1812-699 


0-1019602 


13 45 28-4 


23 


Thalia . 






2-6282154 


1556291 


0-2321817 


10 13 13-1 


24 


Themis . 






3-1439415 


2036-147 


0-1170627 


48 51-6 


25 


Phoeea 






2-3999822 


1358033 


0-2547900 


21 34 54-9 


26 


Proserpina 






2-6552100 


1580-319 


0-0871572 


3 35 38-0 


27 


Euterpe 






2-3463460 


1312-760 


0-1736210 


1 35 30-3 


28 


Bellona 






2-7784700 


1691-986 


0-1500986 


9 21 26-3 


29 


Amphitrite 






2-5536647 


1490-542 


0-0737850 


6 7 49-8 



894 



APPENDIX 



30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60- 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 

80 



Urania 

Euphrosyne 

Pomona 

Polyhymnia 

Circe . 

Leucolhea 

Atalanta 

Fides . 

Leda 

Laatita . 

Harmonia 

Daphne 

Isis . 

Aiiadne 

Nysa . 

Eugenia 

Hestia . 

Aglaia . 

Doris . 

Pales . ■ 

Virginia 

Nemausa 

Euro pa 

Calypso 

Alexandra 

Pandora 

Melete . 

Mnemosyne 

Concordia 

Olympia 

Echo 

Denae . 

Erato . 

Ausonia 

Angelina 

Maximiliana 

Mai a 

Asia 

Leto 

Hesperia 

Panopea 

Niobe . 

Feronia 

Clyde . 

G-alatea 

Eurydice 

Freia . 

Frigga . 

Diana . 

Eurynome 

Sappho 



2-3654133 
3-1571421 

2-5887137 

2-8650620 

2-6870950 

3-0060:548 

2-7479006 

2-6421873 

2-7401483 

2-7699820 

2-2677226 

2-7665526 

2-439999 

2-203358 

2-424741 

2-720510 

2-526151 

2-880197 

3-109420 

3-080259 

2-649741 

2-365579 

3-099773 

2-620957 

2-714864 

2-759621 

2-597110 

3-157288 

2-694441 

2-713870 

2-393088 

3-003978 

3-130695 

2-393785 

2-680557 

3-419842 

2-663544 

2-420277 

2-722658 

2-994932 

2-629115 

2-756159 

2-274958 

2-665541 

2-575212 

2-665856 

3-188960 

2-673752 

2-626316 

2-444173 

2-296375 



1328-795 

2048-988 

1521-332 

1771-328 

1608-877 

1903-680 

1663-792 

1568-674 

1656-758 

1683-862 

1247-335 

1680-725 

1392-14 

1194-61 

1379-10 

1638-98 

1466-52 

1785-38 

2002-71 

1974-60 

1575-45 

1328-94 

1993-39 

1549-84 

1633-88 

1674-45 

1528-74 

2049-13 

1615-48 

1632-98 

1352-19 

1901-71 

2023-29 

1352-78 

1603-01 

2309-98 

1587-77 

1375-29 

1640-92 

1893-12 

1557-09 

1671-30 

1253-31 

1589-56 

1509-45 

1589-84 

2080-04 

1596-91 

1554-60 

1395-71 

1271-05 



0-1276142 

0-2181494 

0-0802144 

0-3381960 

0-1056142 

0.2136315 

0-2971309 

0-1748927 

0-1554322 

0-1111018 

0-0463143 

0-2690810 

0-2256143 

0-1675933 

0-1503545 

0-0892361 

0-1646660 

0-1324005 

0-0769467 

0-2385024 

0-2873290 

0-0662400 

0-1014745 

0-2038207 

0-1966174 

0-1420334 

0-2370434 

01041157 

0-0401686 

0-1173539 

0-1847386 

0-1684354 

0-1709820 

0-1247228 

0-1290952 

01201719 

0-1339165 

0-1848482 

0-1697847 

0-1745242 

0-1950190 

0-1737289 

0-1164641 

0-0454381 

0-2554310 

0-3054377 

0-0302110 

0-1358136 

0-2066804 

0-1953329 

0-2002178 



i 


o i a 


2 6 1-8 


26 26 0-0 


5 28 51-6 


1 56 18-9 


5 26 37-5 


3 10 32-1 


18 41 51-6 


3 7 10-5 


6 58 25-3 


10 20 58-3 


4 15 52-3 


16 3 13-7 


8 34 29-8 


3 27 39-1 


3 41 40-9 


6 34 57-0 


2 17 36-7 


5 9-6 


6 29 40-2 


3 8 45-1 


2 47 36-9 


3 47 53-0 


7 24 40-3 


5 6 42-7 


11 47 11-3 


7 13 28-1 


8 1 50-8 


15 8 1-6 


5 1 6-7 


8 37 41-3 


3 34 18-7 



18 16 32 
2 
5 



9 
12 25-9 
46 54-5 



1 19 51-5 

3 28 9-5 

3 2 24-6 

5 59 27-3 

8 10 16-5 

8 28 25-0 



31 56-5 

18 29-5 

25 55-2 

24 57-6 

15 43-8 

59 S-8 

13 3-0 

27 55-2 

8 39 47-0 

4 36 50-5 

8 36 47-9 



APPENDIX 



895 







a 


P 


e 


i 












O , 1, 


81 


Terpsichore . 


2-S50428 


1757-77 


0-2100461 


7 55 22-0 


82 


Aicmene . 


2 -76414-4 


1678-57 


2299055 


2 50 34 


V 


J.ipiter . 


5*2027760 


4332-5848212 


0-0481626 


1 18 513 


b 


Saturn . 


9-5387861 


10759-2198174 


0-0561502 


2 29 35-7 


w 


Uranus . . 


191823900 


30686-8208296 


0-0466683 


46 28-4 


tp 


Neptune . 


30 05660 


60186-6385 


0-0084962 


1 47 0-6 







SYNOPTIC TABLE OF ELEMENTS (Continued) 




© 


& 


to 


L 


E 


o 


1 


" 


o 


/ 


ti 


° 


' 


" 




d 




45 


57 


30-9 


74 


21 


46-9 


166 





48-6 


1801. Jan. 


1-0, G. 


5 


74 


54 


12-9 


128 


43 


53-1 


11 


33 


3-0 


Do. 




e 








99 


30 


5-0 


100 


39 


10-2 


Do. 




s 


48 





3-5 


332 


23 


56-6 


64 


22 


55-5 


Do. 




i 


80 


48 


47-9 


148 


27 


14-1 


25 


44 


50-6 


1864. Oct. 


12-0, G. 


2 


172 


42 


15-1 


122 


4 


48-1 


354 


44 


38-9 


1864. Sept. 


6-0, G. 


3 


170 


52 


4-9 


54 


44 


9-6 


268 


18 


46-8 


1864. June, 


8-0, G. 


4 


103 


23 


11-7 


250 


38 


10-3 


52 


34 


2-6 


1863. Nov. 


23-0, G. 


5 


141 


25 


43-5 


135 


20 


26-3 


262 


24 


22-5 


1864. June, 


26-0, G. 


6 


138 


37 


21-4 


15 


22 


12-6 


269 


8 


16-9 


1862. May, 


31-0, B. 


7 


259 


48 


13-3 


41 


19 


33 5 


8 


9 


6-5 


1862. Sept. 


12-0, W. 


8 


110 


17 


48-6 


32 


54 


28-3 


68 


48 


31-9 


1848. Jan. 


1-0, B. 


9 


68 


31 


58-4 


71 


32 


19-0 


248 


1 


48-7 


1363. May, 


30-0, B. 


10 


286 


43 


55-0 


234 


49 


57-7 


316 


54 


16-7 


1862. Apr. 


26-0, G. 


11 


125 


5 


56-1 


317 


21 


4-7 


304 


57 


4-5 


1862. July, 


21-0, B. 


12 


235 


33 


34-9 


301 


56 


36-0 


312 


33 


52-4 


1857. Aug. 


2-0, G. 


13 


43 


19 


45-4 


119 





16-2 


111 


51 


59-4 


1864. Jan. 


lo-o, P. 


14 


86 


42 


8-8 


180 


6 


33-9 


98 


57 


33-4 


1862. June, 


1-0, G. 


15 


293 


58 


20-7 


27 


37 


24-7 


107 





24-5 


1862. Jan. 


30-0, W. 


16 


150 


34 


1-6 


14 


44 


57-6 


226 


7 


32-6 


1863. Apr. 


29-5, W. 


17 


125 


21 


37-2 


260 


40 


8-1 


245 


5 


1-1 


1860. July, 


120, B. 


18 


150 


4 


30-7 


15 


9 


58-0 


167 


39 


20-9 


1858. Mar. 


6-0, G. 


19 


211 


27 


18-2 


30 


29 


32-3 


41 


51 


13-5 


1860. Nov. 


8-0, B. 


20 


206 


42 


308 


98 


26 


22-4 


351 


2 


39-8 


1863. Aug. 


29-5, B. 


21 


80 


30 


56-2 


326 


27 


0-6 


349 


21 


33-0 


1860. Jan. 


19 0, G. 


22 


66 


36 


21-8 


56 


34 


13-1 


224 


46 


26-6 


1860. Jan. 


0-0, B. 


23 


67 


39 


12-0 


124 


9 


6-7 


141 


3 


59-3 


1862. Feb. 


19-0, B. 


24 


36 


11 


48-1 


139 


49 


24-2 


62 


24 


39-7 


1862. Oct. 


23-0, G. 


25 


214 


3 


o-i 


302 


53 


583 


108 


4 


27-7 


1863. Jan. 


13-0, B. 


26 


45 


55 


3-4 


234 


50 


23-9 


61 


14 


30-0 


1860. Feb. 


6-0, G. 


27 


93 


46 


22-1 


87 


40 


8-9 


311 


56 


4-5 


1863. July, 


23-0, B. 


28 


144 


41 


9-9 


122 


55 


29-6 


66 


3 


57-4 


1862. Mar. 


24-0, B. 


29 


356 


28 


37-9 


57 


23 


12-1 


283 


40 


58-5 


1863. June, 


30-0, B. 


30 


308 


16 


8-4 


30 


52 


49-6 


73 


59 


9-5 


1862. Dec. 


17-0, B. 


31 


31 


28 


40-0 


94 


15 


6-4 


155 


10 


34-1 


1862. Mar. 


9-0, B. 


32 


220 


47 


37-3 


194 


2 


32-3 


6 


29 


1-5 


1862. Oct. 


1-0. W. 


33 


9 


6 


441 


342 


27 


53-9 


285 


39 


12-6 


1863. May, 


28-5, W. 


34 


184 


42 


23-0 


150 


14 


45-2 


105 


34 


22-0 


1863. Jan. 


3-0, B. 



896 



APPENDIX 



SYNOPTIC TABLE OF ELEMENTS {Continued). 





£ 


« 


L 


E 





' 


it 


o 


' 


it 


° 


' 


" 






d 


35 


355 


51 


209 


201 


26 


27-1 


69 


52 


4-3 


1863. 


Nov. 


16 5, B. 


36 


359 


10 


46 4 


42 


38 


20-0 


71 


20 


46-7 


1861. 


Jan. 





B. 


37 


8 


9 


37-4 


66 


4 


28-2 


42 


34 


35-2 


1856. 


Jan. 


o-o 


B. 


3S 


296 


27 


34-9 


100 


51 


44-3 


112 


58 


27 6 


1856. 


Jan. 


o-o 


B. 


39 


157 


20 


6-6 


2 


11 


36-7 


146 


41 


34-5 


1856. 


Jan. 


1-0 


B. 


40 


93 


34 


23-7 


1 


2 


41-7 


225. 


47 


29-8 


1863. 


May, 


12-0 


B. 


41 


179 


2 


30-1 


220 


5 


20-1 


335 


3 


21-7 


1862. 


Sept. 


23 43E 


,B. 


42 


84 


31 


6-9 


317 


59 


39-2 


247 


46 


19-5 


1860. 


Jan. 


10 


B. 


43 


264 


35 


52-7 


277 


50 


50-4 


132 


1 


30-2 


1863. 


Jan. 


o-o 


B. 


44 


131 


3 


5-2 


111 


28 


29-3 


116 


18 


19-1 


1860. 


Jan. 


28-0 


B. 


45 


148 


5 


51-8 


229 


51 


52-7 


294 


35 


2-8 


1858. 


Jan. 


o-o 


B. 


46 


181 


33 


41-1 


354 


43 


56-3 


86 


7 


34-3 


1863. 


Jan. 


o-o 


B. 


47 


4 


11 


52-2 


313 


53 


0-8 


116 


34 


11-4 


1859. 


June, 


17-0 


B. 


48 


185 


14 


13-2 


76 


52 


38-5 


16 


6 


47-0 


1858. 


Feb. 


3-0 


B. 


49 


290 


29 


4-2 


32 


31 


40-6 


159 


34 


55-3 


1860. 


Jan. 


28-0 


B. 


50 


173 


36 


12-2 


9 


51 


303 


89 


16 


52-2 


1863. 


Jan. 


o-o 


B. 


51 


175 


39 


11-2 


175 


10 


53-1 


177 


11 


25-3 


1858. 


Mar. 


25-5 


B. 


52 


129 


57 


16-5 


101 


54 


57-2 


136 


20 


51-4 


1858. 


Jan. 





B. 


53 


144 


1 


52-5 


92 


47 


35-8 


239 


19 


168 


1863. 


May, 


31-5 


B. 


54 


313 


49 


7 


294 


53 


461 


149 


40 


21 -6 


1861. 


Jan. 


8-0 


B. 


55 


10 


57 


29-1 


11 


28 


38-6 


28 


26 


57-6 


1858. 


Dec. 


30-0 


B. 


56 


194 


25 


59-4 


293 


37 


37-9 


62 


16 


45 7 


1862. 


Dec. 


18-0 


B. 


57 


200 


5 


25-1 


52 


53 


13-0 


2S 


35 


25-6 


1860. 


Jan. 


1-0 


B. 


58 


161 


11 


39-8 


180 


17 


24-0 


162 


28 


26-1 


1860. 


Jan. 


o-o 


B. 


59 


170 


22 


31-4 


16 


54 


44-4 


190 


6 


18-6 


1863. 


Jan. 


o-o 


B. 


60 


191 


59 


47-3 


98 


30 


17-3 


232 


27 


11 8 


1863. 


Jan. 


o-o 


B. 


61 


334 


16 


57-9 


342 


44 


12-7 


73 


6 


25-4 


1862. 


Jan. 


o-o 


B. 


62 


126 


12 


55 • 1 


34 





8-2 


176 


1 


23 2 


1863. 


Mar. 


25-5 


B. 


63 


338 


3 


48-9 


269 


41 


07 


177 


7 


9-1 


1861. 


Mar. 


5-0 


B. 


64 


311 


4 


46*9 


123 


43 


50-3 


182 


57 


5-4 


1861. 


May, 


28-0 


B. 


65 


158 


53 


48-7 


258 


22 


17-1 


192 


17 


21-4 


1861. 


Mar. 


18-0 


B. 


66 


8 


13 


12-5 


38 


13 


4-7 


188 


42 


325 


1861. 


May, 


27-0 


W. 


67 


202 


40 


10-1 


306 


18 


47-9 


313 


38 


23-3 


1862. 


Jan. 


o-o 


B. 


68 


44 


37 


547 


358 


57 


32 


282 


33 


47 2 


1862. 


Jan. 


00 


B. 


69 


186 


59 


4 


111 


8 


42-0 


164 


2 


21-0 


1861. 


June, 


3-0 


B. 


70 


48 


16 


27-8 


299 


47 


31-6 


250 


34 


164 


1861. 


June, 


00 


B. 


71 


316 


18 


43-4 


221 


58 


46-8 


322 


15 


20-S 


1861. 


Sept. 


25-5 


B. 


72 


207 


37 


13-1 


309 


48 


379 


22 


52 


55-9 


1863. 


Jan. 


o-o 


B. 


73 


7 


32 


18-9 


61 


33 


51 -1 


1S9 


47 


11 9 


1862. 


May, 


26-0 


W. 


74 


200 


29 


20-2 


6 


42 


365 





31 


29-4 


1862. 


Sept. 


16-0 


B. 


75 


359 


52 


19-1 


3. > 4 


40 


12 1 


4 


40 


23-1 


1862 


Nov. 


00 


W. 


76 


212 


29 


32 5 


67 


10 


17-9 


28 


48 


2-8 


1862. 


Oct. 


24-5 


0. 


77 


2 


7 


1-7 


58 


9 


1-3 


39 


25 


26-4 


1863. 


Jan. 


o-o 


B. 


78 


334 


2 


342 


121 


13 


r>9-2 


173 


41 


43-8 


1863. 


May, 


8-5 


, B. 


79 


206 


42 


40-0 


44 


20 


33-1 


45 


50 


13 3 


1864. 


Jan. 


10 


O. 


80 


218 


31 


18-8 


355 


7 


20-7 


61 


25 


323 


1865. 


Dec. 


3-0 


. B. 


81 


2 


31 


45-1 


48 


17 


29-6 


29 


35 


16-4 


1864. 


Nov. 


13-0 


, B. 



APPENDIX 

SYNOPTIC TABLE OF ELEMENTS {Continued). 



897 







Q 


u 


L 


E 




o 


/ // 


o 


' 


n 


o 


/ n 




d 


82 


26 


50 33 


131 


12 


48 


94 


4 41 


1865. Jan. 


0-0, B. 


% 


93 


26 18-9 


11 


8 


34-6 


112 


15 230 


1801. Jan. 


] -0, G. 


b 


111 


56 37-4 


89 


9 


29-8 


135 


20 6-5 


Do. 




W 


72 


59 35 3 


167 


31 


16-1 


177 


48 23-0 


Do. 




q> 


130 


7 31-9 


43 


17 


30-3 


335 


6 0-4 


1850. Jan. 


1-0, G. 



SYNOPTIC TABLE OF ELEMENTS (Continued). 

• See 357 a, Note F. 





M 


D 


d 


A 


T 


7 


H 


e 






a 




h 


m s 





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882000 


1923-64 


0-25 


609 


7 


82 


45 — 


45940 




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4865751 


3183 


6-89 


1-14 


21 


5 — 






6-674 




* 


401839 


8108 


17-55 


0-84 


23 


21 — 




• 


1-911 




e 


359551 


7925-648 




1-00 


23 


56 4-1 


66 


32 6 


1-000 


298 


a 


2680337 


4546 


6-46 


0-72 


24 


37 22-7 


59 


41 49 


0-431 


62 


% 


1047-871 


90734 


37-91 


0-24 


9 


55 21-3 


86 


54 30 


0-036 


16-84 


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3501-600 


76791 


17-50 


o-ii 


10 


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61 


49 — 


o-oii 




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20470 


35307 


3-91 


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2-88 


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o-oo 





898 



APPENDIX 



NAMES OF DISCOVERERS AND DATES OF DISCOVERY 
OF THE ASTEROIDS 



Ceres 


Piazzi 


Jan. 1, 1801- 


Pallas 


Olbers 


Mar. 28. 1802. 


Juuo 


Harding 


Sept. 1, 1804. 


Vesta 


Olbers 


Mar. 29, 1807. 


Astraea 


Hencke 


Dec. 8, 1845. 


Hebe 


Hencke 


July 1, 1847. 


Iris 


Hind 


Aug. 13, 1847. 


Flora 


Hind 


Oct. 18, 1847. 


Metis? 


Graham 


Apr. 25, 1848. 


Hygeia 


Gasparis 


Apr. 12, 1849. 


Parthenope 


Gasparis 


May 11, 1850. 


Victoria 


Hind 


Sept. 13, 1850. 


Egeria? 


Gasparis 


Nov. 12, 1850. 


Irene 


Hind 


May 19. 1851. 


Eunomia 


Gasparis 


July 29, 1851. 


Psvche 


Gasparis 


Mar. 17. 1852. 


Thetis 


Luther 


Apr. 17, 1852. 


Melpomene 


Hind 


June 24, 1852. 


Foriuna 


Hind 


Aug. 22, 1852. 


Massilia 


Gasparis 


Sept. 19, 1852. 


Lutetia 


Golds chmidt 


Nov. 15, 1852. 


Calliope 


Hind 


Nov. 16, 1852. 


Thalia 


Hind 


Dec. 15, 1852. 


Themis? 


Gasparis 


Apr. 5, 1853. 


Phocea 


Chacornac 


Apr. 6, 1853. 


Proserpina 


Luther 


May 5, 1853. 


Euterpe 


Hind 


Nov. 8, 1853. 


Bellona 


Luther 


Mar. 1, 1854. 


Amphitrite 


j Marth 
I Pogson 


Mar. 1, 1854. 
Mar. 1, 1854. 


Urania 


Hind 


July 22, 1854. 


Euphrosyne 


Ferguson 


Sept. 1, 1854. 


Pomona 


Gold schmidt 


Oct. 26, 1854. 


Polyhymnia 


Chacornac 


Oct. 28, 1854. 


Circe 


Chacornac 


Apr. 6. 1854. 


Leucothea 


Luther 


Apr. 19, 1855. 


Atalanta 


Gold schmidt 


Oct. 5, 1855. 


Fides 


Luther 


Oct. 5, 1855. 


Leda 


Chacornac 


Jan. 12, 1856. 


Lsetitia? 


Chacornac 


Feb. 8, 1856. 


Harmonia 


Goldschmidt 


Mar. 1, 1856. 


Daphne 


. Goldschmidt 


May 22. 1856. 


Isis? 


Pogson 


Mav 23, 1856. 


Ariadne 


Pogson 


Apr. 15, 1857. 


Nysa 


Goldschmidt 


Mav 27, 1857. - 


Eugenia? 


Goldschmidt 


June 28, 1857. 


Hestia? 


Pogson 


Aug. 16, 1857. 


Aglaia 


Luther 


Sept. 15, 1857. 


Doris 


Goldschmidt 


Sept. 19, 1857. 



APPENDIX 



899 



Pales? 


Goldschraidt 


Sept. 19, 1857. 


Virginia 


Ferguson 


Oct 4, 1857. 


Nernuasa? 


Laurent 


Jan. 22, 1858. 


Europa 


Goldschmidt 


Feb. 4, 1858. 


Calypso 


Luther 


Apr. 4, 1S58. 


Alexandra 


Goldschmidt 


Apr. 11, 1858. 


Pandora 


Searle 


Sept, 10, 1858. 


Melete 


Goldschraidt 


Sept. 9, 1857. 


Mnemosyne 


Luther 


Sept. 22, 1S59. 


Concordia 


Luther 


Mar. 24, 1860. 


Olvmpia 


Chacornac 


Sept. 13, 1860. 


Echo 


Ferguson 


Sept. 15, 1860. 


Danae 


Goldschmidt 


Sept. 9, 1860. 


Erato 


Lesser 


Sept. 14, 1860. 


Ausonia 


Gaspari3 


Feb. 11, 1861. 


Angelina 


Tempel 


Mar. 6. 1861. 


Maximi liana 


Tempel 


Mar. 10. 1861. 


Maia 


Tattle 


Apr. 10, 1861. 


Asia 


Pogson 


Apr. 18. 1861. 


Leto 


Luther 


Anr. 29, 1861. 


Hesperia 


Schiaparelli 


Apr. 29, 1861. 


Panopea 


Goldschmidt 


May 5, 1861. 


Niobe 


Luther 


Aug. 13. 1861. 


Peronia 


Peters 


Jan. 29. 1862. 


Clytie 


T uttle 


Apr. 7. 1862. 


Galatea 


Tempel 


Auo-. 29, 1862. 


Eurydice 


Peters 


Sept. 22, 1862. 


Freia 


D' Arrest 


Nov. 14, 1862. 


Frigga 


Peters 


Nov. 12, 1862 


Diana 


Luther 


Mar. 15, 1863. 


Eurynome 


Watson 


Sept. 14. 1863. 


Sappho 


Pogson 


Mar 3, 1864. 


Terpischore 


Tempel 


Sept. 30. 1864. 


Alcmene 


Luther 


Nov. 27, 1864. 



Note. — Many of the names of the Asteroids appear to us very unhappily 
chosen. Thus, confusion is very likely to arise in printing or speaking, be- 
tween Iris and Tsis, Lutetia and Lsetitia, Thetis and Metis, Thetis and Themis.. 
Yesta and Hestia, Hygeia and Egeria, Egeria and Eugenia, Pallas and Pales. 
Is it too much to hope that the discoverers of the interfering members of these 
pairs will reconsider their names? It is not yet too late: the Nymphs, Dryads 
Oceanidae, etc., afford an infinite choice of classic names, graceful and euphoni- 
ous. Metis i3 known to few as a mythological name, Pales to fewer as that of 
a female divinity, Nemausa to none as the name of anybody (the ancient name 
of Nismes was Nemans-Ms). 



900 APPENDIX 



III 

SYNOPTIC TABLE OF THE ELEMENTS OF THE ORBITS 
OF THE SATELLITES, SO FAR AS THEY ARE KNOWN ■ 

1. THE MOON 

Mean distance from the earth 60r27343300 

Mean sidereal revolution 27d3216614l8 

Mean synodical ditto 29d'530588715 

Excentricity of orbit......... 0'0549080 ! 70 

Mean revolution of nodes.... .6793d*39l080 

Mean revolution of apogee 3232d'575343 

Mean longitude of npde at epoch 13° 53' 17"* 1 ? 

Mean longitude of perigee at ditto .....266 .10 7 *5 

Mean inclination of orbit 5 8 39 -98 

Mean longitude of moon at epoch .118 17 8 *3 

Mass, that of the earth being 1, 0-011364 

Diameter in miles 2164*6 

Density, that of the earth being 1, 0*55654 

1 The distances are expressed In equatorial radii of the primaries. The 
epoch is Jan. 1, 1801, unless otherwise expressed. The periods, etc., are ex- 
pressed in mean solar days. 



APPENDIX 



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yo4 



APPENDIX 



Elements of the Orbit of the Earth as computed by M. Leverrier for 
Intervals of 10,000 Years, from 100,000 Years before a.d. 1800 to 
100,000 YEARS after THAT date. (Connaissance des Temps pour Tan 1843.) 



Epoch. 


Excentricity. 


Longitude of 

Perihelion. 


Inclination. 


Longitud 


eof SI 


— 100,000 


00473 


316 


18 


3 


45 


31 


96 


i 

34 


— 90,000 


0-0452 


340 


2 


2 


42 


19 


76 


17 


— 80,000 


0-0398 


4 


13 


1 


18 


58 


73 


47 


— 70,000 


0316 


27 


22 


1 


13 


5S 


136 


8 


— 60,000 


0-0218 


46 


8 


2 


36 


42 


136 


29 


— 50,000 


0-0131 


50 


14 


3 


40 


11 


116 


9 


— 40,000 


0-0109 


28 


36 


4 


3 


1 


91 


59 


— 30,000 


0-0151 


5 


50 


3 


41 


51 


66 


49 


— 20,000 


0-0188 


44 





2 


44 


12 


41 


34 


— 10,000 


0-0187 


78 


28 


1 


24 


35 


16 


39 





0-0168 


99 


30 

















+ 10,000 


0-0155 


134 


14 


1 


14 


26 


148 


15 


-f- 20,000 


0-0047 


192 


22 


2 


7 


46 


124 


29 


4- 30,000 


0-0059 


318 


47 


2 


33 


19 


100 


29 


-f 40,000 


0-0124 


6 


25 


2 


27 


53 


75 


31 


4- 50,000 


0173 


38 


3 


1 


51 


54 


48 


13 


4- 60,000 


0-0199 


64 


31 





51 


52 


10 


47 


4- 70,000 


0-0211 


71 


7 





34 


35 


220 


38 


-J- 80,000 


0-0188 


101 


38 


1 


45 


40 


170 


15 


4- 90,000 


0-0176 


109 


19 


2 


40 


56 


139 


3 


4- 100,000 


0-0189 


114 


5 


3 


2 


57 


109 


57 



Excentricity and Longitude of the Perihelion of the Earth's Orbit 
for a Million of Years Past and to Come, as computed by Mr. Croll 
(Phil. Mag. April, 1866) 



Number of i 
Years before Excentricity. 
Epocli 1800. 



000,000 
950,000 
900.000 
850,000 
800,000 
750,000 
700,000 
650,000 
600,000 
550,000 
500.000 
450,000 
400,000 
350,000 
300,000 
250,000 
200,000 
150,000 
100,000 
50,000 
0,000 



0-0151 
0-0517 
0-0102 
0-0747 
0-0132 
0-0575 
0-0220 
0-0226 
0-0417 
0-0106 
0-0388 
0-0308 
0-0170 
0-0195 
0-0424 
0-0258 
0-0569 
0-0332 
0-0473 
0-0131 
0-0168 



Luigitu 


de of 


Perihelion. 


248 


22 


97 


51 


135 


2 


239 


28 


343 


49 


27 


18 


208 


13 


141 


29 


32 


34 


251 


50 


193 


56 


356 


52 


290 


7 


182 


50 


23 


29 


59 


39 


168 


18 


242 


56 


316 


2 


50 


3 


99 


30 



Number of 
Years after 
Epoch 1800. 



50,000 
100,000 
150,000 
200,000 
250.000 
300,000 
350,000 
400,000 
450,000 
500,000 
550,000 
600,000 
650,000 
700,000 
750,000 
800,000 
850,000 
900,000 
950,000 
1,000,000 



Excentricity, 



0-0173 
0-0191 
0-0353 
0-0076 
0-0286 
0-0158 
0-0098 
0-0429 
0-0231 
0-0534 
0-0259 
0-0395 
00169 
0-0375 
0-0195 
0-0639 
0-0144 
0-0659 
0-0086 
0-0528 



Longiti 


ade of 


Perihelion. 


o 


/ 


38 


12 


114 


50 


201 


57 


303 


30 


350 


54 


179 


29 


201 


40 


6 


9 


98 


37 


157 


26 


287 


31 


285 


43 


144 


3 


17 


12 





53 


140 


38 


176 


41 


291 


16 


115 


13 


57 


21 



APPENDIX 



E 


LEMENTS OF i 


ASTEROID* 


5 DISCO 


VERED Sit 


VCl 


1864 


83 


I! oat rice . . 


2-43.7244 


1389-78 


0-0859251 


o 

5 





3-0 


84 


Clio . . 






2-361823 


1325-77 


0-2361542 


9 


22 


25-5 


86 


Io . . 






2-683884 


1579-14 


0-1908907 


11 


53 


9-7 


86 


Semelo . 






3-090786 


1984-73 


0-2049113 


4 


47 


44-6 


87 


Sylvia . 






3-494110 


2385-66 


0-0823511 


10 


50 


56-9 


S3 


Thisbe . 






2-768885 


1682-89 


0-1651097 


5 


14 


3.0-1 


♦ 89 


Julia 






2-549835 


1487-19 


0-1803041 


16 


11 


25-5 


90 


Antiope 






3-138789 


2031-15 


0-1747166 


2 


16 


47-8 


91 


iEgina . 






2-491717 


1436-63 


0-0662013 


2 


9 


31-6 


92 


Undina . 






3-191618 


2082-65 


0-1031710 


9 


57 


3-0 


93 


Minerva 






2-755910 


1671-07 


0-1402670 


8 


36 


31-8 


94 


Aurora . 






3-160410 


2052-12 


0-0889312 


8 


5 


18-5 


95 


Arethusa 






3-068786 


1963-58 


0-1465338 


12 


51 


1-5 


96 


J£gle . 






3-054500 


1949-71 


0-1402763 


16 


6 


31-1 


97 


Clotho . 






2-66S621 


1592-31 


0-2568941 


11 


44 


58-4 


98 

99 

100 


Ianthe . 






2-684535 


1606-58 


0-1891891 


15 


42 


35-1 


Hecate . 






2-993973 


1892-21 


0-1690456 


6 


9 


50*4 


101 


Helena . 






2-573119 


1507-61 


0-1394040 


10 


4 


19-5 


102 


Miriam . 






2-662367 


1586-72 


0-2547668 


5 


6 


3-3 


103 


Hera 






2-702265 


1622-52 


0-0806707 


5 


21 


35-2 


104 


Clyrnene 






3-179809 


2071-09 


0-1973404 


2 


53 


26-7 


105 


Artemis 






2-379975 . 


1341-06 


0-1761972 


21 


38 


59-0 


106 


Dione . 






3-201010 


2091-84 


0-1950237 


4 


41 


33-2 





SI 


CO 


L 


E 





1 


" 





1 


a 





1 


11 






83 


27 


33 


31-2 


193 


49 


32-0 


339 


49 


35-1 


1866. Oct. 


1-5, B. 


84 


337 


22 


1-1 


339 


11 


58-1 


353 


48 


43-6 


1865. Nov. 


13-0, B. 


85 


203 


53 


27-0 


322 


35 


52-7 


102 


36 


43-6 


1867. Jan. 


0-0, B. 


86 


87 


55 


52-1 


28 


39 


6-7 


39 


12 


57-1 


1866. Jan. 


20-0, B. 


87 


76 


24 


4-6 


336 


59 


9-2 


251 


32 


58-3 


1866. May 


16-42, B. 


88 


277 


42 


52-5 


308 


50 


52-4 


304 


59 


21-7 


1866. Aug. 


4-5, B. 


89 


311 


30 


11-7 


353 


17 


8-1 


330 


44 


57-1 


1866. Aug. 


31-0, B. 


90 


71 


14 


59-6 


301 


1 


28-5 


351 


44 


29-6 


1866. Oct. 


28-0, B. 


91 


11 


41 


34-8 


68 


54 


3-3 


281 


27 


18-8 


1869. June 30-0, B. 


92 


102 


52 


32-7 


333 


21 


56-0 


278 


25 


58-8 


1867. Jan. 


0-0, B. 


93 


5 


4 


11-4 


275 


38 


16-3 


342 


40 


15-4 


1867. Oct, 


2-0, B. 


94 


4 


34 


36-4 


44 


37 


19-5 


159 


47 


3-2 


1870. Jan. 


0-0, B. 


95 


244 


22 


31-4 


30 


22 


34-4 


127 


59 


11-2 


1869. Jan. 


0-0, B. 


96 


322 


51 


4-3 


164 


16 


51-5 


275 


11 


5-1 


1870. Jan. 


0-0, B. 


97 


160 


36 


34-8 


65 


33 


36-0 


126 


8 


50-0 


1868. Jan. 


0-0, B. 


98 

99 

100 


354 


16 


43-2 


147 


43 


7-5 


150 


25 


16-9 


1868. Jan. 


0-0, B. 


128 


16 


59-3 


343 





46-1 


316 


7 


52-7 


1868. July. 


11-0, B. 


101 


343 


35 


o-i 


328 


40 


51-0 


346 


33 


18-1 


1868. Sept. 


14-0, B. 


102 


211 


32 


43-2 


355 


9 


10-2 


303 


28 


54-0 


1868. Jan. 


0-0, B. 


103 


135 


56 


56-4 


326 


15 


4-8 





2 


16-1 


1868. Oct. 


13-0, B. 


104 


43 


46 


42-1 


62 


11 


55-4 


18 


24 


14-2 


1868. Sept. 


14-0, B. 


105 


187 


54 


1-8 


242 


36 


17-8 


348 


54 


8-0 


1868. Oct. 


13-0, B. 


106 


62 


42 


38.9 


35 


37 


53-6 


22 


25 


o-? 


1868. Oct. 


11-0. B. 



906 APPENDIX 

NAMES OF DISCOVERERS AND DATES OF DISCOVERS 



83 


Beatrice 


Gasparis 


April 26, 1865. 


84 


Clio 


Luther 


Aug. 25, 1865. 


85 


Io 


Peters 


Sept. 19, 1865. 


86 


Seraele 


Tietjen 


Jan. 4, 1866. 


87 


Sylvia 


Pogson 


May 16, 1866. 


88 


Thisbe 


Peters 


June 15, 1866. 


89 


Julia 


Stepban 


Aug. 6, 1866. 


90 


Antiope 


Lullier 


Oct. 1, 1866. 


91 


iEgina 


Stephan 


Nov. 4, 1866. 


92 


Unclina 


Peters 


July 26, 1867. 


93 


Minerva 


■Watson 


Aug. 24, 1867. 


94 


Aurora 


Watson 


Sept. 6, 1867. 


95 


Arethusa 


Luther 


Nov. 23, 1867. 


96 


JE<r\e 


Coggia 


Peb. 17, 1868. 


97 


Olotho 


Tern pel 


Feb. 17, 1868. 


98 


Ian the 


Peters 


April 18, 1868. 


99 




Borelly 


May 28, 1868. 


100 


Hecate 


Watson 


July 11, 1868. 


101 


Helena 


Watson 


Aug. 15, 1868. 


102 


Miriam 


Peters 


Aug. 23, 1868. 


103 


Hera 


Watson 


Sept. 7, 1868. 


104 


Clymene 


Watson 


Sept. 13, 1868. 


105 


Artemis 


Watson 


Sept. 16, 1868. 


106 


Dione 


Watson 


Oct. 10, 1868. 



APPENDIX 



907 



TABLE OF NUMBERS IN FREQUENT USE AMONG 

ASTRONOMERS (See Note F)* 





Number or 


Logarithm 


Log. 


Circumference of a circle to diameter 1 tt 


Multiplier 




Reciprocal 


3-1415927 


0-4971499 


9-5028501 


No. of degrees in circular are= radius . 


57-2957795 


1-7581225 


8-2418775 


No. of seconds in circular arc .... 


206264-8 


5-3144251 


4-6S55749 


X"o. of seconds in the whole circumference, 








360° 


1296000 


6-1126050 


3-8873950 


X~o. of sine of 1' to radius =1 . . . 


0-0002909 


6-4637261 


3-5302739 


No. whose natural logarithm is 1 e 


2-7182818 


0-4342945 


9-5657055 


Multiplier to reduce common to natural 








logarithms 


2-3025851 


0-3622149 


9-6377851 


Amplitude of the probability curve for 








probability =$ 


0-47694 


9-6963040 


0-3036960 


Multiplier to reduce French metres to 








British feet 


3-28090 


0-5159929 


9-4840071 


Multiplier to reduce French metres to 








British inches 


39-37079 


1-5951741 


8-4048259 


Multiplier to reduce French toises to 








British feet 


6-394593 


0-8058129 


9-1941871 


Multiplier to reduce French grammes to 








British grains 


15-423460 


1-1884351 


8-8115649 


Multiplier to reduce French litres to 








British cubic inches 


61-027043 


1-7855223 


8-2144777 


Length of seconds pendulum, London, 








vacuum, sea-level, in inches . 


39-13929 


1-5926129 


8-4073871 










gravity in 1" (lat. 35° 16') . . . 


32-18169 


1-5076088 


8*4923912 


Earth's mean diameter in British standard 








miles 


7912-410 


3-8983088 


6-1016912 


Mean barometric pressure at sea-level on 








1 square inch in lbs 


14-7304 


1-1682145 


8-8317855 


Weight of 1 cubic inch of distilled water, 








62° Fahr., bar. 30 inches, in grains 


252-458 


2-4021892 


7-5978108 


SpeciQc gravity of mercury at 32° Fahr., 








(water at 40°) 


13-596 


1-1334112 


8-8665888 


Velocity of sound in dry air at 32° Fahr., 








in feet per second 


1089-42 


3-0371964 


6-9628036 


Telocity of light in vacuo in British stand- 








ard miles per second 


191515* 


5-2822029 


4-7177971 


Multiplier to reduce sidereal days to mean 








solar days ........ 


0-9972696 


9-9988126 


0-0011874 


Multiplier to reduce sidereal year to mean 








solar days 


365-2563612 


2-5625977 


7-4374023 


Multiplier to reduce tropical year to mean 








solar days ........ 


365-2422414 


2-5625809 


7-4374191 


Multiplier to reduce mean synodic lunar 








months to mean solar days . 


29-5305887 


1-4702726 


8-5297274 


The sun's mean equatorial horizontal par- 








allax 


g". 577(3* 


0-9333658 


9-0666342 



908 APPENDIX 

Table of Numbers in frequent use among Astronomers (Continued) 



The moon's mean equatorial horizontal 
parallax 

The sun's mean apparent semidiameter . 

The moon's mean apparent semidiameter 

Constant of aberration 

Maximum of nutation of obliquity of ecliptic 

Maximum of nutation in longitude 

Mean annual precession of the equinoxes 
(for 1790) 

Constant of refraction (=ref. at 45° alt, 
bar. 29-6, therm. 50° Fahr.) . . . 

Mean horizontal refraction 

Mean obliquity of the ecliptic, Jan. 1, 1860 



Number or 
Multiplier 



3422"'325 
961 // -820 
934"-685 
20"-4452 
9"'2236 
17"-2524 

50"-23492 

57"-524 

1980" 

23° 27' 27"'38 



Logarithm 



5343212 
9830938 
9706652 
3105914 
■9649005 
2368495 



1-7010057 

1-7598491 
3-2966652 



Log. 
Reciprocal 



6-4656788 
7-0169062 
7-0293348 
8-6894086 
9-0350995 
8-7631505 

8-2989943 

8-2401509 
6-7033348 



INDEX 



N.B. The references are to the articles, not to the pages. 

... attached to a reference Dumber indicates that the reference extends to the 
article cited, and several subsequent in succession 



Aberration of light explained, 329. 
Its uranograpliical effects, 333. Of 
an object in motion, 335. How dis- 
tinguished from parallax, 805. Sys- 
tematic, 862. 

Abonl Wefa, 705. 

Acceleration, secular, of moon's mean 
motion, 740. 

Adams, 506, 767. 

Adjustment, errors of, in instruments, 
136. Of particular instruments. (See 
those instruments.) 

^Itna, portion of earth visible from, 32. 
Height of, 32 note. 

Agathocles, eclipse of, 933 b, and 
Table of Eras, art. 926. 

Air, rarefaction of, 33. Law of den- 
sity, 37. Refractive power affected 
by moisture, 41. 

Airy (Gr. B., Esq.), his results respect- 
ing figure of the earth, 220. Re- 
searches on perturbations of the 
earth by Venus, 726. Rectification 
of the mass of Jupiter, 757. 

Albategnius, 430. 

Algol, 821. 

Altitude, used to find time, 129. 

rind azimuth instrument, 187. 

— — s, equal, method of, 188. 



Andromeda, nebula in, 874. 

Angle of position, 204. Of situation, 
311. 

Angles, measurement of, 163, 167. 
Hour, 107. 

Angular velocity, law of, variation of, 
350. 

Annular nebulas, 875. 

Anomalistic year, 384. 

Anomaly of a planet, 499. 

Antarctic circle, 364, 382. 

Apex of aberration, 343. Of parallax, 
?43. Of refraction, 343. Solar, 854. 
Of shooting stars, 902, 904. 

Aphelion, 368. 

Apogee, 368. Of moon, 406. Period 
of its revolution, 687. 

Apsides, 406. Revolution of lunar, 
409. Motion of, investigated, 675. 
Application to lunar, 676... Mo- 
tion of, illustrated by experiment 
692. Of planetary orbits, 694. Li- 
bra tion of, 694. Motion in orbits 
very near to circles, 696. In ex- 
centric orbits, 697... 

Arago, 387, 395, 432, 776 j, 877. 

Arcs of meridian, how measured, 211. 
Measures of some, 216. Russian, 
Indian and Freneh, 223 a, Note D. 

Arctic circle, 364, 382. 

Areas, Kepler's law of, 352..., 490. 



(909) 



010 



INDEX 



Argelander, his researches on variable 
stars, 820..., on sun's proper mo- 
tion, 854. 

Argo, nebulae in, 887. Irregular star v, 
in constellation, 830, 896 g, Note K. 

Ariel, 551. 

Arislillus, 430. 

Ascension, right., 108. (See Right 
ascension.) 

Asteroids, their existence suspected 
previous to their discovery, 505. 
Appearance in telescopes, 525. 
Gravity on surface of, 525. Ele- 
ments, Appendix, Synoptic Table. 
Total mass of, inconsiderable, 525. 
List of discoverers, and dates of 
discovery. (See Synoptic Tables.) 

Astrsea, discovery of, 505. 

Astrometer, 783. 

Astronomy. Etymology, 11. General 
notions, 11. 

Atmosphere, constitution of, 33... 
Possible limit of, 36. Its waves, 

37. Strata, 37. Causes refraction, 

38. Twilight, 44. Total mass of, 
242. Of Jupiter, 513. Of the sun, 
see Sun. Of the moon, 431. 

Attraction of a sphere, 446-450. (See 
Gravitation.) To a spheroid, 238. 

Augmentation of moon's apparent 
diameter, 404. 

Augustus, his reformation of mistakes 
in the Julian calendar, 919. Era 
of, 926. 

Aurora Eorealis, 115. 

Australia, excessive summer tempera- 
ture of, 369. 

Auzout, 158 note. 

Averages of results, their use, 137. 

Axis of the earth, 82. Rotation per- 
manent, 56. Major, of the earth's 
orbit, 373. Of sun's rotation, 392. 
Polar, its exact length, 223 a, 
Note D. Equatorial, longer, ib. 



Axis of a planetary oi bit. Momentary 
variation of, caused by the tangen- 
tial force only, 658, 660. Its varia- 
tions periodical, 661... Invariable 
ity of, and how understood, 668. 

Azimuth, 103. 

and altitude instrument, 187. 



Babinet, his torsion gravimeter, 
Note E. 

Bail}', his observation of annular 
eclipse of the sun, 425. His 
beads, 425. 

Barometer, nature of its indication, 
33. Use in calculating refraction, 
43. In determining heights. 287. 

Base, measurement of, 273. 

Beads, Baily's, 425. 

Beer and Maedler, their work on the 
moon, 429. 

Belts of Jupiter, 512. Of Saturn, 514. 

Bessel, his results respecting the lig- 
ure of the earth, 220. Discovers 
parallax of 61 Cygni, 812. 

Biela's comet, 579... 

Biot, his aeronautic ascent, 32. 

Birt, his examination of lunar craters, 
430 a, Note H. 

Bissextile, 932. Omar's proposal for 
its periodical omission, Note A on 
art. 926. 

Bode, his (so-called) law of planetary 
distances, 505. Violated in the case 
of Neptune, 507. 

Boguslawski, remarkable observation 
of Halley's comet by, 571 note. 

Bohnenberger, his principle of collima- 
tion, 179. 

Bond, Prof., his observations of inte- 
rior ring of Saturn, 521. His dis- 
covery of an eighth satellite of 
Saturn, 548. 



INDEX 



911 



Borda, his principle of repetition, 198. 

Bouvard, his suspicion of extraneous 
iutiuence on Uranus, 760. 

Brewster (Sir D.), his polarizing eye- 
piece, 204 d. 

British geometric system of weights 
and measures, 223 a, Note D. 

Broun, his torsion gravimeter, Note E. 



C-ESar, his reform of the Romau cal- 
endar, 917. 

Calendar, Julian, 917. Gregorian, 914. 

Calms, equatorial, 244. 

Carrington, solar phenomenon ob- 
served by, 387 c, Note G-. His 
researches on the sun's spots, ib. 

Cause and effect, 439, and note. 

Cavendish, his experiment, 776 h. 

Centre of the earth, 80. Of the sun, 
462. 

of gravity, 360. Revolution 

about, 452. Of the earth and moon, 
452. Of the sun and earth, 451. 

Centrifugal force. Elliptic form of 
earth produced by, 224. Illus- 
trated, 225. Compared with grav- 
ity, 229. Of a body revolving on 
the earth's surface, 452. 

Ceres, discovery of, 505. 

Challis (Prof.), 506 note. 

Charts, celestial, 111. Construction of, 
291. Bremiker's, 506, and note. 

Chinese records of comets, 574. Of 
irregular stars, 831. 

Chronometers, how used for determin- 
ing differences of longitude, 255. 

Circle, arctic and antarctic, 95. Ver- 
tical. 100. Hour, 106. Divided, 163. 
Meridian, 174. Reflecting. 197. Re- 
Galactic, 793. Mu- 
Arctic and antarctic, 
-'a co-ordinate, 181. 



peating, 198 
ral, 163, 168 
364. 312. - 
Clamu, 163. 



Clarke (Alvan), bis detection of a com- 
panion of Sirius, 859 a, Note J. 

Clarke (Uapi. A. K.), his computation 
of the dimensions and figure of the 
earth, 223 a, Note D. 

Clausen, his orbit of comet of 1843, 
596. 

Clepsydra, 150. 

Climate, 366... Secular changes of, 
369 6..., 701. 

CIoca, 151. En or and rate of, how 
found, 253. 

Clockwork applied to equatorial, 186. 

Clouds, greatest height of, 34. Magel- 
lanic, 892. . . Solar, Note O, 395 d d. 

Clusters of stars, 864... Globular, 867. 
Irregular, 869. 

Collimation, line of, 185. 

Collimator, floating, 178. Bohnenber- 
ger's, 179. 

Colored stars, 851... 

Colson, his maps, 2S4, 

Colures, 307. 

Comets, 554. Seen in day-time, 555, 
590. Tails of, 556. ..566, 599. Ex- 
treme tenuity of, 558. General 
description of, 560 Motions of, 
and described, 561... Parabolic, 
564. Elliptic, 567... Hyperbolic, 
564. Dimensions of. 565. Of Hal- 
ley, 567... Of Caesar, 573. Of 
Encke, 576. Of Biela. 579. Its 
subdivision into two. 580. Of Faye, 
584. Of Lexell, 585. Of De Yico, 
586. Of Brorsen. 587. Of Peters, 
588. Synops's of elements (Appen- 
dix). Increase of visible dimensions 
in receding from the sun, 571, 580. 
Great, of 1843, 589... Its supposed 
identity with many others. 594... 
Inteiest attached to subject, 597. 
Cometary statistics, and conclusions 
therefrom, 601. Conclusions from 
the phenomena of their tails, 570. 



912 



INDEX 



Possible cause of their acceleration 
of period, 570. Heat sustained by, 
592. Principal discoverers of, 597. 
Periodic, form two distinct families, 
601 a. Of 1680, 573. Of 1556, 
574. Orbits of, identical with those 
of meteors, 905 i i, j j, Note N. 

Commensurability (near) of mean mo- 
tions; of Saturn's satellites, 550. 
Of Uranus and Neptune, 669, and 
note. Of Jupiter and Saturn, 720. 
Earth and Venus, 728. Effects of, 
719. 

Compensation of disturbances, how 
effected, 719, 725. 

Compression of terrestrial spheroid, 221 

Configurations, inequalities depending 
on, 655... 

Conjunctions, superior and inferior, 
469, 473. Perturbations chiefly 
produced at, 713. 

Consciousness of effort when force 
is exerted, 439. 

Constellations, 60, 301. How brought 
into view by change of latitude, 52. 
Rising and setting of, 58. 

Copernican explanation of diurnal mo- 
tion, 76. Of apparent motions of 
sun and planets, 77. 

Correction of astronomical observa- 
tions, 324... — s. Uranographical 
summary, view of, 342... Order 
of application of, 345. 

Craters, lunar, their great size ex- 
plained, 430 a, Note H. Models 
of by Mr. Nasmyth, 437. 

Culminations, 125. Upper and lower, 
126. 

Cycle, of conjunctions of disturbing 
and disturbed planets, 719. Me- 
tonic, 926. Callippic, ib. Solar, 
921. Lunar, 922. Of indictions, 
923. Of eclipses, 426. 

Cyclones, 245 a. 



Darkening glasses, 204 e. 

Dates, Julian and Gregorian, interval 
between, how computed, 927... 

Dawes (Rev. W. R.), his mode of 
observing solar spots, 204 e. His 
discovery of the sun's interior en- 
velope, 389 a. His observation of 
interior ring of Saturn, 521. His 
discovery of bright spots on Jupi- 
ter's belts, 512. Of an eighth satel- 
lite of Saturn, 548. 

Day, solar, lunar, and sidereal, 143. 
Ratio of sidereal to solar, 305, 909, 
911. Solar unequal, 146. Mean 
ditto, invariable, 908. Civil and as- 
tronomical, 147. Intercalary, 916. 

Days elapsed between principal chron- 
ological eras, 926. Rules for reck- 
oning between given dates, 927. 
Formula for, Note C. Of week 
not the same over the gJobe, 257. 
And nights, their inequality ex- 
plained, 365. 

Declination, 105. How obtained, 295. 
Parallels of, 113. 

Definitions, 82. 

De la Rue (W., Esq.), his stereographs 
of the moon, 430 a, Note H. His 
photographs of the sun, 400 a. 
His verification of the willow-leaved 
structure of its surface, 387 a, 
Note G. 

Degree of meridian, how measured, 
210... Error admissible in, 215. 
Length of in various latitudes, 
216, 221. 

Density of earth, how determined, 
776 a b. 

Densities of sun and planets, 508 b. 
(See also Synoptic Table.) 

Diameters of the earth, 220, 221. Of 
planets, synopsis, Appendix. (See 
also each planet.) 



INDEX 



913 



Dilatation of comets in receding from 
the sun, 578. 

Diminution of gravity at equator, 231. 

Dione, 548. 

Dip of horizon, 23. 

sector, 18, 19. 

Disks of stars, SI 6. 

Distance of the moon, 403: the sun, 
357; fixed stars, 807, 812...; of 
the earth and planets from the sun, 
hitherto overrated, correction re- 
quired, 357 a, Note F; polar, 105. 

Districts, natural, in heavens, 302. 

Disturbing forces, 455. Of sun, on 
moon, 606. Nature of, 609... Gen- 
eral estimation of, 611. Numerical 
values, 612. Unresolved in direc- 
tion, 614. Resolution of, in two 
modes, 615, 618. Effects of each 
resolved portion, 616... On moon, 
expressions of, 676. Geometrical 
representations of, 676, 717. 

Diurnal motion explained, 58. Paral- 
lax, 339. Rotation, 144. 

Double refraction, 202. Image mi- 
crometer, a new, described, 203. 
Comet, 530. Nebulae, 878. 

Double Stars, 833... Specimens of 
each class, 836. Orbital motion 
of, 839. Subject to Newtonian 
attraction, 843. Orbits of particu- 
lar, 843. Dimensions of these or- 
bits, 844, 848. Colored, 851... Ap- 
parent periods affected by motion 
of light, 863. 

Dove, his law of temperature, 370. 
Of rotation of winds. 245 a. 

Drainage basins, 289. 



Eart;I. Its motion admissible, 15. 
Spherical form of, 18, 22..., 419. 
Optical effect of its curvature, 25. 
Diurnal rotation of, 52. Uniform, 



56. Permanence of its axis, 57. 
Figure spheroidal, 207, 219... Di- 
mensions of, 209, 210, 220, 233 a, 
Note D. Elliptic figure a result of 
theory, 229. Temperature of sur- 
face, how maintained, 336. Ap- 
pearance as seen from moon, 436. 
Velocity in its orbit, 474. Disturb- 
ance by Venus, 726. Density of, 
776, a b. Soiid content of, 776 I. 

Eclipses, 411... Solar, 420, Note 0. 
Lunar, 421... Annular, 425. Peri- 
odic return of, 426. Number possi- 
ble in a year, 426. Of Jupiter's 
satellites, .266, 538. Of Saturn's, 
549. Total of sun, phenomena of, 
395. Ancient, their use in fixing 
dates, 933... 

Ecliptic, 305... Its plane slowly va- 
riable, 306. Cause of this varia- 
tion explained, 640. Poles of, 307. 
Limits, solar, 412. Lunar, 527. 
Obliquity of, 305, 306. 

Egyptians, ancient, chronology, 912. 

Elements of a planet's orbit, 493. 
Variations of, 652... Of double 
star orbits, 843. Synoptic table 
of planetary, etc., Appendix. 

Ellipse, variable, of a planet, 653. 
Momentary or osculating, 654. 

Elliptic motion a consequence of gravi- 
tation, 446. Laws of, 489... Their 
theoretical explanation, 491. 

Ellipticity of the Earth, 208, 221. Of 
its meridians, 223 a, Note D. Of its 
equator, ib. Of Jupiter, 512 a. 

Elongation, 331. Greatest, of Mer- 
cury and Venus, 467. 

Enceladus, 548 note. 

Encke, comet of, 576. His hypothesis 
of the resistance of the ether, 
577. 

Epoch, one of the elements of a plan- 
et's orbit, 496. Its variation not 



914 



INDEX 



independent, T30. Variations inci- 
dent on, 731 744. 

Equation of light, 335. Of the centre, 
375. Of time, 379. Lunar, 452. 
Annual, of the moon, 738. Men- 
strual, of the sun, 528. 

Equator, 84, 112. Of the earth some- 
what elliptic, 223 a, Note D. 

Equatorial, 185. 

instrument, 84. 112. 

calms, 224. 

Equilibrium, figure of, in a rotating 
body, 224. 

Exemplified by an experiment, 225. 

Equinoctial, 97, 113. Time, 148, 935. 

Equinox, 293, 303, 307, 362. 

Equinoxes, precession of, 318. Its 
effects, 313. In what consisting, 
314... Its physical cause explained, 
642... 

Eras, chronological list of, 926. 

Erratic stars, 297. 

Errors, classification of, 133. Instru- 
mental, 135... Of adjustment, 136. 
Their detection, 140. Destruction 
of accidental ones by taking means, 
137. Of clock, how obtained, 293. 

Establishment of a port, 754. 

Ether, resistance of, 577. 

Evection of moon, 748. 

Evolute of ellipse, 219, 220. 

Excentricities, stability of Lagrange's 
theorem respecting, 701. 

Excentricity in a divided circle, how 
eliminated, 141. Earth's orbit, 354. 
Pas* and future, see tables in Ap- 
pendix. How ascertained, 377. Of 
the moon's, 405. Momentary per- 
turbation of, investigated, 670. Ap- 
plication to iunar theory, 688. Yari- 
ations of, in orbits nearly circular, 
696. In excentric orbits, 697. Per- 
manent inequalities depending on, 
719. 



Facul^e of the sun, 388. Explained, 
395 a. 

Fuye, comet of, 584 and Appendix. 

Field of view, 160. 

Fixed stars. See Stars. 

Fizeau, his measure of the velocity of 
light, 545. 

Flora, discovery of, 505. 

Focus, upper. Its momentary change 
of place, 670, 671. Path of, in vir- 
tue of both elements of disturbing 
force, 704. Traced in the case of 
the moon's variation, 706... And 
parallactic inequality, 712. Circu- 
lation of, about a mean situation m 
planetary perturbations, 727. 

Force, metaphysical conception of, 
439. 

Forced vibration, principle of, 650. 

Forces, disturbing. See Disturbing 
Force. 

Foncault, his pendulum experiment, 
245 c. His gyroscope, 245 h. His 
determination of the velocity of 
light, 357 a, Note F. 

Friction as possibly a source of the 
sun's heat, 400, 905 a..., 387 c c, 
Note G. 

G 

Galactic circle, 793. Polar dis- 
tance, ib. 

Galaxy composed of stars, 302. Sir 
W. Herschei's conception of its 
form and structure, 786. Distri- 
bution of stars generally referable 
to it, 786. Its course among the 
constellations, 787... Difficulty of 
conceiving its real form, 792. Tele- 
scopic analysis of, 797. In some 
directions unfathomable, in others 
not, 798. 

Galle (Dr.), 506. Finds Neptune in 



INDEX 



915 



place indicated by theory, 768. 
First notices the interior ring of 
Saturn, 522. 

Galloway, his researches on the sun's 
proper motion, 855. 

Gaseoigne, 158. 

Gaspans (Sig. De) discovers a new 
planet (Appendix). 

Gauging the heavens, 793. 

Geocentric longitude, 503. Place, 371, 
497. 

Geodesical measurements — their na- 
ture, 247. • 

Geography, 111, 129... 205... 

Glaisher, his balloon ascent, 32. 

Glasses, darkening, 204 c. 

Globe, artificial, an experiment with, 
245 j. 

Globular clusters, 865. Their dynami- 
cal stability, 866. Specimen list of, 
867. 

Golden number, 922. 

Goodricke, his discovery of variable 
stars, 821. 

Goldschmidt. (See Asteroids.) His ob- 
servations of Sirius, 859 a, Note J. 

Gravitation, how deduced from phe- 
nomena, 444... Elliptic motion a 
consequence of, 490... 

Gravity, centre of. See Centre of 
Gravity. 

Gravity diminished by centrifugal 
force, 231. Measures of, statical, 
234. Dynamical, 235. Force of, 
on the moon, 433... On bodies at 
surface of the sun, 440. Of other 
planets, see their names. 

Greenwich, latitude of, 123. 

Gregorian reform of calendar, 915. 

Gyroscope, 245 h... 



Habit ability of the moon, 436 a b. 
Hadley, his sextant, 194. His expla- 



nation of the trade winds. (See 
Winds.) 

Halley. His comet, 567. Notices 
acceleration of moon's motion, 740. 
First notices proper motions of the 
stars, 852, 

Hansen. His detection of long in- 
equalities in the moon's motions, 
745... 

Harding discovers Juno, 505. 

Harvest moon, 428 b. 

Heat, supply of, from sun alike in 
summer and winter, 368. How 
affected by changes in the earth's 
orbit, 369 b,.. How kept up, 400. 
Sun's expenditure of, estimated, 
397. Received from the sun by 
different planets, 508. Endured 
by comets in perihelio, 592. 

Hebe, discovery of, 505. 

Heights above the sea, how meas- 
ured, 286. Mean of the continents, 
289. 

Heliocentric place, 372, 498, 500. 

Heliometer, 201. 

Helioscope, 204 e. 

Hemispheres, terrestrial and aqueous, 
284. 

Hencke discovers Hebe and Astrsea, 
505. 

Henderson, his determination of the 
parallax of « Centauri, 807. 

Herschel (Sir ¥m.), discovers Uranus, 
595, and two satellites of Saturn, 
548. His methodj of gauging the 
heavens, 793. Yiews of the struc- 
ture of the Milky Way, 786. Of 
nebular subsidence, and sidereal 
aggregation, 869, 874. His cata- 
logues of double stars, 835. Dis- 
covery of their binary connection, 
839. Of the sun's proper motion, 
854. Classifications of nebulas, 868, 
879, note. 

Astronomy— Vol. XX— 20 



910 



INDEX 



Herschel (Miss C), comets discovered 
by, 597. Nebula discovered by, 
874. 

Herschel (Prof. A. S.), his obser- 
vations of meteors, Note N, 905 
a a, j j. 

Herschel (Lieut. J.), his observations 
of nebulae, Note K, 896 g, 896 h h; 
of solar eclipses, Note 0. 

Hind, his calculation of the return of 
comets, 574. Classification of com- 
ets, 601. 

Hipparchus, 281. 

Hodgson, his observations of a phe- 
nomenon in the sun, 387 a, Note G. 

Horizon, 22. Dip of, 23, 195. Ra- 
tional and sensible, 74. Celestial, 
98, 113. Artificial, 173. 

Horizontal point of a mural circle, how 
determined, 175. 

Horrockes, 158. 

Hour circles, 113, 136. Glass, 150. 

Huggins (W., Esq.), his observations 
on new star in Corona, 831 ; on the 
spectra of nebulae, Note K, 896 h h; 
of the red prominences projecting 
from the sun, Note O; of the tails 
of comets, Note O, 395 d d; oi 
Sirius, Note J, 859 b b. 

Humboldt, his determination of the 
mean heights of continents, 289. 

Hurricanes, 245 b. 

Hyperion, 518. 



Iapetus, 548. 

Illumination of field of view, 204 a. 

Red, its advantages, 204 b. Of 

wires, 204 b. 
Immersions and emersions of Jupiter's 

satellites, 538... 
Inch, British, remarkable relation of 

to the length of the earth's axis 

of rotation, 223 a, Note D. 



Inclination of the moon's orbit, 406. 
Of planets' orbits disturbed by or- 
thogonal force, 619. Physical im- 
portance of, as an element, 632. 
Momentary variation of, estimated, 
633. Criterion of momentary in- 
crease or diminution, 635. Its 
changes periodical and self-cor- 
recting, 636. Application to case 
of the moon, 638. 

Inclinations, stability of, Lagrange's 
theorem, 639. Analogous in their 
perturbations to excentricities, 699. 

Indictions, 923. 

Inequality. Parallactic of moon, 712. 
Great, of Jupiter and Saturn, 720... 

Inequalities, independent of excentric- 
ity, theory of, 702... Dependent 
on, 719. 

Instrument -making, its difficulties, 
131. Equatorial, 185. Alt-azimuth, 
187. 

Instrumental errors, how detected, 
139... 

Instruments, theory of, 140... 

Intercalation, 916. 

Iris, discovery of, 505... 

Iron, meteoric, 888. 

Isis. (See table of Asteroids.) 



James (Sir H.), his projection of the 
sphere, 283. His measure of attrac- 
tion of Arthur's Seat, 776 e. 

Julian period, 924. Date, 930. Ref- 
ormation, 918. 

Juno, discovery of, 505. 

Jupiter, physical appearance and de- 
scription of, 511. Ellipticity of, 
512. Belts of, 512. Gravity on 
surface, 508. Satellites of, 510, 
535. Their use for determining 
longitudes, 266. Seen without sat. 
ellites, 543. Density of, 508 b. 



INDEX 



917 



Recommended as a photometric 
indud, 783. Klenients of, etc. 
(See Synoptic Table, Appendix.) 
Jupiter and Saturn, their mutual per- 
turbations, 700, 720... 



Sater, his mode of measuring small 
intervals of time, 150. His colli- 
mator, 178. 

Kepler, his laws, 352, 439, 487. Their 
physical interpretation, 490... 



Lagging of tides, 753. 

Lagrange, his theorems respecting the 
stability of the planetary system, 
639, 669, 701. 

Laplace accounts for the secular ac- 
celeration of the moon, 740. 

Larissa, eclipse of, 933. And Table, 
926. 

Lasseil, his observation of spots on 
Jupiter's belts, 512. His discovery 
of Ariel and Umbriel, 557. Of an 
eighth satellite of Saturn, Hyperion, 
548. His observations at Malta, 551. 

Latitude, terrestrial, 88. Parallels of, 
89. How ascertained, 119, 129. Ho- 
mer's mode of obtaining, 248. On 
a spheroid, 247. Celestial, 308. 
Heliocentric, how calculated, 500. 
Geocentric, 503. Of a place inva- 
riable, 31. Of Greenwich, 123. 

Laws of nature, how arrived at, 139. 
Subordinate, appear first in form of 
errors, 139. Kepler's, 352, 487... 

Leap-year, 917. 

Level, spirit, 176. Lines, 289. Sea, 
285. Strata, 287. 

Leverrier, 506, 507, 767. 

Lexell, comet of, 585. 

Libration of the moon, 435. How 



availed of for stereoscopy, 430 a, 
Note H. Of apsides, 694. 

Light, aberration of, 331. Velocity 
of, 331. How ascertained, 545. 
Overrated, its correction, 357 a, 
Note F. Equation of, 335. Ex- 
tinction of, in traversing space, 
798. Distance measured by its mo- 
tion, 802... Of certain stars com- 
pared with the sun, 817... Effect 
of its motion in altering apparent 
period of a double star, 863. Zodi- 
acal, 897. 

Limits, ecliptic. (See Ecliptic limits.) 

Local time, 252. 

Lockyer (J. N., Esq.), his spectro- 
scopic observations, Note O. 

Lohrmann, his charts of the moon, 437. 

London, centre of the terrestrial hemi- 
sphere, 284. 

Longitude, terrestrial, 90. How de- 
termined, 121, 251... By chronom- 
eters, 255. By signals, 264. By 
electric telegraph, 262. By shoot- 
ing stars, 265. By Jupiter's satel- 
lites, etc., 266. By lunar observa- 
tions, 267... Celestial, 308. Mean 
and true, 375. Heliocentric, 500, 
Geocentric, 372, 503. Of Jupiter's 
satellites, curious relations of, 542. 

Lunar distances, 367... Volcanoes 
and craters, 430 a, Note H. 

Lunation (synodic revolution of the 
moon), its duration, 418. 

m 

Maclear, his measurement of arc at 

the Cape, 220. His rediscovery of 

d' Arrest's comet, 600 b. 
Magellanic clouds, 892... 
Magnetic storm, remarkable, 387 a, 

Note G. 
Magnetism, terrestrial, connected with 

spots on the sun, 394 c. 



918 



INDEX 



Magnitudes of stars, 780... Common 
and photometric scales of, 780... 
And Appendix. 

Main, his observation of Saturn, 522 a. 

Maps, geographical, construction of, 
273, Celestial, 290... Of the moon, 
437. Projections used in, 246, 280... 

Mark, meridian, 190. 

Mars, phases of, 484. Gravity on sur- 
face, 508. Continents and seas of, 
510. Elements (Appendix). Rota- 
tion on its axis, 510. 

Maskelyne, his measure of attraction 
of a mountain, 776 e. 

Misses of planets determined by their 
satellites, 532. By their mutual 
perturbations, 757. Of Jupiter's 
satellites, 758. Of the moon, 759. 
Of the sun and planets, overesti- 
mated, correction, 357 a, Note F. 

Mean motions of Jupiter's satellites 
singular relation of, 542. Do., of 
Saturn's, 550. 

Menstrual equation, 528. 

Mercator's projections, 281, 283. 

Mercury, synodic revolution of, 472. 
Telocity in orbits, 474. Stationary 
points of, 476. Phases, 477. Great- 
est elongations, 482. Transits of, 
483. Heat received from sun, 508. 
Physical appearance and descrip- 
tion, 509. Elements of (Appendix). 

Meridian, terrestrial, 85. Celestial, 
101. Line, 87, 190. Circle, 174. 
Mark, 190. Arc, how measured, 
210. Arcs, lengths of, in various 
latitudes, 216. Length of a degree 
of, in feet, 221. 

Messier, his catalogue of nebula?, 865. 

Meteors, 898. Periodical, 900... 
Heights of, 904. Recent discov- 
eries respecting, Note N. Of Aug. 
10 and Nov. 13, their orbits, 905 
i i, j j, Note N. 



Metis, discovery of. 505. 

Michell, his invention of the torsion- 
balance, 776 h. Its application to 
measure density of the earth, 776 i. 
His speculations on the distribution 
of stars, 833. 

Micrometers, 199... Double image, 
200... Position, 204. 

Milky way. (See Galaxy, 302.) 

Mimas, 550, and note. 

Mines, oscillations of pendulum in, 
833. 

Mira Ceti, 820. 

Mitchell (Miss), her discovery of a 
comet, 558, 597. 

Month, lunar, 418, 934, note. 

Moon, her motion among the stars, 
401. Distance of, 403. Magnitude 
and horizontal parallax, 404. Aug- 
mentation, 404. Her orbit, 405. 
Revolution of nodes, 407. Apsides, 
409. Occultation of stars by, 414. 
Phases of, 416. Brightness of sur- 
face, 417, note. Redness in eclipses, 
422. Sidereal and synodic, revolu- 
tion, 418. Physical constitution of, 
429... Destitute of sensible atmos- 
pere, 431. Mountains of, 430. Vol- 
canic craters of, 430 a, Note H. 
Climate, 431... Inhabitants, 434. 
Habitabiiity, 436 a, b. Libration, 
435. Visible in total eclipse, 424 
note. Harvest, 428 b. Influence 
on weather, 432 and note. Rota- 
tion on axis, 435. Appearance from 
earth, 436. Maps and models of 
437, 430 a. Stereographic repre- 
sentation of, ib. Real form of orbit 
round the sun, 452. Gravity on 
surface, 508. Her mass and den- 
sity. (See Synoptic Table of Ele- 
ments, III.) Motion of her nodes 
and change of inclination explained, 
638... Motion of apsides, 6^6.,, 



INDEX 



919 



Variation of excentricity, 688... 
Parallactic inequality, 712. An- 
nual equation, 738. Evection, 
748. Variation, 705... Tides pro- 
duced by, 751. 

Motion, apparent and real, 15. Diur- 
nal, 52. Parallactic, 68. Relative 
and absolute, 78... Angular, how- 
measured, 149. Proper, of stars, 
852... Of sun, 854. 

Mountains, their proportion to the 
globe, 29. Of the moon, 430. At- 
traction of, 776 d. 

Mowna Roa, 32. 

Mural circle, 168. 



N 



Nabonassar, era of, 226. 

Nadir, 99. 

Nasmyth (Jas. Esq.), his discovery of 
the willow-leaved structure of the 
sun's photosphere, 387 a, Note G- 
His model of lunar craters, Note I 

Nebulas, classifications of, 868, 879 
note. Law of distribution, 868. Re 
solvable, 870. Elliptic, 873. Of An 
dromecia, 874. Annular, 875. Plan 
etary, 876. Colored, ib. Double 
878. Of sub-regular forms, 881 
882. Irregular, 883. Of Orion 
885. Of Argo, 887. Of Sagitta 
rins, 888. Of Cygnus, 891. Miss 
ing, 896 a, Note K. Variable, ib 
General catalogue of, ib. Spectro 
scopic observations of, Note K. 

Nebular hypothesis, 872. 

Nebulous matter, 871. Stars, 880. 

Neptune, discovery of, 506, 768. Per- 
turbations produced on Uranus by, 
analyzed, 765... Place indicated by 
theory, 767. Elements of, 771... 
Perturbing forces of, on Uranus, 
geometrically exhibited, 773. Their 



effects, 774... Its periodic time, 
774. 

Newton, his theory of gravitation, 
490... el passim. 

Nodes of the sun's equator, 390. Of 
the moon's orbit, 407. Passage of 
planets through, 460, Of planetary 
orbits, 495. Perturbation of, 620... 
Criterion of their advance or recess, 
622. Recede on the disturbing or- 
bit, 624... Motion of the moon's, 
theory of, 638. Analogy of their 
variations to those of perihelia, 699. 

Nomenclature of Saturn's satellites, 
548, note. 

Nonagesimal point, how found, 310. 

Noon, 'mean and apparent, 378. 

Normal disturbing force and its effects, 
618. Action on excentricity and 
perihelion, 673. Action on lunar 
apsides, 676. Of Neptune on Ura- 
nus, its effects, 775. 

Nubecula?, major and minor, 892... 

Number, golden, 922. 

Nutation, in what consisting, 321. 
Period, 322. Common to all celes- 
tial bodies, 323. Explained on 
physical principles, 648. Mode of 
correcting for, 325. 

O 

Oberon, 551. 

Object glass, divided, 201. 

Objects, test for telescopes, 836. 

Obliquity of ecliptic, 303. Produces 
the variations of season, 362. Slow- 
ly diminishing, and why, 640. 

Observation, astronomical, its pecu- 
liarities, 138. 

Occultation, perpetual, circle of, 113. 
Of a star by the moon, 413... Of 
Jupiter's satellites by the body. 541. 
Of Saturn's, 549. 



920 



INDEX 



Olbers discovers Pallas and V'esta, 
505. His h} T potliesis of the partial 
opacity of space, 798. 

Omar, his proposal for a rule for bis- 
sextiles, Note A, 926. 

Opacity, partial, of space, 798. 

Oscillations, forced principle of, 650. 

Orbits of planets, their elements (Ap- 
pendix) of double stars, 843. Of 
comets. (See Comets.) 

Orthogonal disturbing force, and its 
effects, 616, 619. 

Orthographic projection, 280. 



Palitzch discovers the variability of 
Algol, 821. 

Pallas, discovery of, 505. 

Palm trees, their disappearance from 
Judea, 369 c. 

Parallactic instrument, 185. Inequal- 
ily of the moon, 712. Of planets, 
713. Unit of sidereal distances, 
804. Motion, 68. 

Parallax, 70. Geocentric or diurnal, 
339. Heliocentric, 341. Horizon- 
tal, 355. Of the moon, 404. Of 
the sun, 357, 479, 481. Overesti- 
mated, its probable correction, 357 a, 
Note F. Of Mars, ib. Annual, of 
stars, 800. How investigated, 805... 
Of particular stars, 812, 813, 815. 
Systematic, 862. Effect of, on lunar 
distances, 271. As a uranographi- 
cal correction, 341. Calculation of, 
338. 

Paris, longitude of, 262. 

Peak of Teneriffe, 32. 

Pendulum-clock, 89. A measure of 
gravity, 235. 

Pendulum used as a measure of grav- 
ity, 235. Seconds, length of, 225. 



Foucault's, 245 e... Used to meas- 
ure density of the earth, 776/, g. 

Penumbra, 420. Of solar spots. (See 
Spots.) 

Perigee, 368 a. Of the moon, 406. 

Perihelia and excentriciiies, theory 
of, 670. 

Perihelion, 368. Of the earth, its 
period of revolution, 366 b. Effect 
of its revolution on seasons, 369 b. 
Longitude of, 495. Passage, 496. 
Heat endured by comets in, 592. 

Period, Julian, 924. Of Planets (Ap.) 

Periodic time of a body revolving at 
the earth's surface, 442. Of plan- 
ets, how ascertained, 486. Law of, 
488. Of a disturbed planet perma- 
nently altered, 734... 

Periodical stars, 820... List of, 825. 

Prospective, celestial, 114. 

Perturbations, 602... Of Uranus by 
Neptune, 767... 

Peters, his researches on parallax, 
815. On proper motions of stars, 
859. 

Phases of the moon explained, 416. 
Of Mercury and Venus, 465, 477. 
Of superior planets, 484. 

Photographic representation of the 
moon, 457. 

Photometric scale of star magnitudes, 
780. 

Photometry of stars, 783. 

Piazzi discovers Ceres, 505. 

Picard, 158. 

Piddington on Cyclones, 245 d. 

Pigott, variable stars discovered by s 
824... 

Places, mean and true, 374. Geo- 
metric and heliocentric, 371, 497. 

Planetary nebulas, 876... Note K. 

Planets, 299, 455. Apparent motions, 
457... Stations and retrogradations, 
459. Reference to sun as their cen- 



INDEX 



921 



tre, 462. Community of nature with 
the earth, 463. Apparent diameters 
of, 464. Phases of, 465. Inferior 
and superior, 476. Transits of (see 
Transit). Motions explained, 468. 
Distances, how concluded, 471. Pe- 
riods, how found, 472. Syuodical 
revolutions. 472, 486. Superior, 
their stations and retrogradations, 
485. Magnitude of orbits, how 
concluded, 485. Elements of, 495. 
(See Appendix for Synoptic Table.) 
Densities, 508. Physical peculiari- 
ties, etc., 509... Illustration of their 
relative sizes and distances, 526. 
Division into classes, 525 a. 

Plautamour, his calculations respect- 
ing the double comet of Biela, 
5S3. 

Pleiades, 865. Assigned by Madler 
as the central point of the sidereal 
system, 861. Bright nebula dis- 
covered in, 896 a, Note K. 

Plumb-line, direction of, 23. Use of, 
in observation, 175. On a spheroid, 
219. 

Pogson, his observation of variable 
stars, 825. Discovery of asteroids. 
(See Synoptic Table.) His observa- 
tion of a temporary star in a globu- 
lar cluster, 896 a, Note K. 

Polar distance, 105. Point, on a mural 
circle, 170, 172. 

Polarization, 387. 

Polarizing eye-piece, 204 d. 

Poles, 83, 112, 113. Of ecliptic, 307. 
Their motion among the stars, 317. 

Pole-star, 95. Useful for finding the 
latitude, 171. Not always the same, 
318. What, at epoch of the build- 
ing of the pyramids, 319. 

Pores of the sun's surface, 387. Ex- 
plained, 398. In what consisting, 
387 a, Note G-. 



Position, angle of, 204. Microme- 
ter, ib. 

Pouillet (M.), his measure of solar 
radiation, 397 a, note. 

Powell (E. B., Esq.) his elements of 
double star orbits, 843. His obser- 
vations of a variable nebula, 896 a, 
Note K. 

Powell (Prof.), his explanation of the 
gyroscope, 345 /. 

Prassepe, Cancri, 865. 

Precession of the equinoxes, 312. 
In what consisting, 314... Effects, 
313. Physical explanation, 642... 

Priming and lagging of tides, 753. 

Principle of areas, 490. Of forced 
vibrations, 650. Of repetition, 198. 
Of conservation of vis viva, 663. 
Of collimation, 178. 

Pritchard (Rev. C), his verification of 
the willow-leaved structure of the 
sun's photosphere, 387 a, Note O. 

Problem of three bodies, 608. 

Problems in plane astronomy, 127... 
309... 

Projection of a star on the moon's 
limb, 414 note. 

Projections of the sphere, 280... A 
simple and convenient described, 
283. Of equal areas, 283 b. 

Proper motions of the stars, 852. Of 
Sirius, inequalities in, 859. Proba- 
ble explanation of, 859 a, Note J. 
Of the sun, 853, 858 a, Note L. 

Pyramids, 319. 

R 

Radial disturbing force, 615... 

Radiation, solar, on planets, 508. On 
comets, 592. 

Rate of clock, how obtained, 293. 

Reading off, methods of, 165. Oppo- 
site effect of the eliminating errors, 
141. 



922 



INDEX 



Redfield on hurricanes, 245 d. 
Reduction of astronomical observa- 
tions, 336. 

Reflecting circle, 197. 

Reflectors, large, how col lima ted, 2 04/. 

Reflection, observations by, 173. 

Reformation of calendar, by Cassar, 
918. By Augustus, 919. By Pope 
Gregory, 932. Proposal by Omar 
for, 906, Note A. 

Refraction, 38. Astronomical and its 
effects, 39, 40. Measure of, and 
law of variation, 43. How detected 
by observation, 142. Terrestrial, 
44. How best investigated, 191. 

Reid (Sir W.), on hurricanes, 245 d. 

Repetition, principle of, 198. 

Resistance of ether, 577. 

Retrogradations of planets, 459. Of 
nodes. (See Nodes.) 

Reversal, principle of, 161. 

Reynaud (M.), his speculations on 
variation of climate, 369 c, note. 

Rhea, 548 note. 

Right ascension, 108. How deter- 
mined, 293. 

Rings of Saturn, dimensions of, 514. 
Phenomena of their disappearance, 
515... Equilibrium of, 518... Mul- 
tiple, 521, and Appendix. Interior, 
521... Appearance of, from Saturn, 
522. Attraction of, on a point 
within, 735 note. 

Rising and setting of celestial objects, 
time of, 128. 

Rittenhouse, his principle of collima- 
tion, 178. 

Rockets used as signals for longitude, 
545. 

Rosse (Earl of), his great reflector, 870, 
882. His account of nebulas, ib. 

Rotation, diurnal, 58. Its effect on 
figure of the earth, 224. Of the 
earth demonstrated, 231... Paral- 



lactic, 68. Of planets, 509... Of 

Jupiter, 512. Of fixed stars on 

their axes, 820. 
Runker (Madame), her discovery of 

a comet, 597. 
Russell, his charts and globe of the 

moon, 437. 



Sadler, two sunrises and sunsets ob- 
served by, in one day, 26. 

Safford (Professor), his researches on 
the proper motion of Sirius, 859 a, 
Note J. 

Saros, 426. 

Satellites of Jupiter, 511. Of Saturn, 
518, 547. Nomenclature of, 548 
note. Remarkable relation of peri- 
ods among, 550. Discovery of an 
eighth, 548. Of Uranus, 523, 552. 
Of Neptune, 524, 553. Used to 
determine masses of their prima- 
ries, 532. Obey Kepler's laws, 
533. Eclipses of Jupiter's, 535... 
Longitude determined by (see Lon- 
gitude). Relations among their mo- 
tions, 542. Other phenomena of, 
540. Their dimensions and masses, 
540. Discovery, 544. Velocity of 
light ascertained from, 545. 

Saturn, remarkable deficiency of den- 
sity, 508. Rings of, 514. (See 
Rings.) Physical description of, 
514. Satellites of, 547 and Ap- 
pendix. (See also elements in 
Appendix.) 

Schehallien, its attraction measured, 
776 e. 

Schmidt, his observations of solar 
eclipses, 395. 

Schubert (Gen.), his determination of 
the dimensions and figure of the 
earth, 223 a, Note D. 



INDEX 



923 



Schwabe, his discovery of periodicity 
of solar spots, 394 a. Of excen- 
tricity of Saturn's rings, 519 note. 

Sou, proportion of its depth to radius 
of the globe, 31. Its action in 
modelling the external form of the 
earth, 227. 

ins explained, 362... Temperature 
of, 366. 

Sector, zenith, 192. 

Secular variations, how detected, 385. 
Explained, 655... 

Selenography, 437. 

Sextant, 193... 

Shadow, dimensions of the earth's, 
422, 428. Cast by Venus, 267. 
Of Jupiter's satellites seen on 
disk, 540. 

Shooting stars, 115. Used for find- 
ing longitudes, 265. Periodical, 
900. (See Meteors.) 

Sidereal time, 110, 143, 910. Tear. 
(See Year.) Day. 144. (See Day.) 

Signals, rocket, 545. Telegraphic, 
259 .. 

Signs of zodiac, 380. 

Sirius, its parallax and absolute light, 
818. Its revolution about an un- 
seen centre, 859. Small compan- 
ions of, 859 a, Note J. Its recess 
from the sun, do., 859 b b. 

Situation, angle of, 311. 

Solar cycle, 921. 

Solstices, 363. 

Space, question as to its absolute 
transparency, 798. 

penetrating power, 803. 

Sphere, 95. Projections of, 280. At- 
traction of, 735 note. 

Spherical excess, 277. 

Spheroid, attraction of, 238. 

Spheroidal form of Earth (see Earth) 
produces inequalities in the moon's 
motion, 749. 



Spots on Sun, 389... Seen with naked 
eye, 387, 394 a. Size of, 386. Na- 
ture of, 389. Movements of, 390. 
Duration of, 394. Periodicity of, 
394 a. Connection with our sea- 
sons, 394 b. 

Spring, proposed use of for determin- 
ing variation of gravity, 274. 

Standards of length, weight, and ca- 
pacity, 223 a, Note D. 

Stars, visible by day, 61. Fixed and 
erratic, 297. Fixed, 777... Their 
apparent magnitudes, 778... Com- 
parison by an astronomer, 783. 
Law of distribution over heavens, 
785... Alike in either hemisphere, 
794. Parallax of certain, 815. 
Disks of, 816. Real size and 
absolute light, 817. Periodical, 
820... Temporary, 827. Irregu- 
lar, 830. Missing, 832. Double, 
833... Colored, 851, and note. 
Proper motions of, 852, Irregu- 
larities in motions accounted for, 
859. Clusters of, 864... Neb- 
ulous, 879... Nebulous-double, 
880. 

Stationary points of planets, 459. 
How determined, 475. Of Mer- 
cury and Tenus, 476. 

Stereograms of the moon, 430 a, 
Note H. 

Stereographic projection, 281. 

Sticklastad, eclipse of, 926 and Table, 
933 c. 

Stones, meteoric, 899. Great shower 
of, ib. 

Strove, his researches on the law of 
distribution of stars, 793. Discov- 
ery of parallax of « Lyrse, 813. 
Catalogue and observations of 
double stars, 835. 

Struve (Otto), his researches on 
proper motions, 854. His conjee- 



924 



INDEX 



ture of the increase in breadth of 
Saturn's ring, 522 a. 

Style, old and new, 932. 

Sun, oval shape and great size on 
horizon explained, 47. Apparent 
motion not uniform, 347. Beams 
converging, 13.5. Orbit elliptic, 349. 
Greatest and least apparent diame- 
ters, 348. Actual distance, 357. 
Overrated, its probable correction, 
357 a, Note F. Magnitude, 358. 
Rotation on axis, 359, 360, Note 
G, 387 c c. Mass, 449. Physical 
constitution, 386. Spots, ib... Its 
parallax, 355. Light, not polar- 
ized, 387 note. Its interior enve- 
lope, 389 a. Its pores, 387, 387 a, 
Note G. "Willow-leaved structure 
of its photosphere, ib. Faculse, 
S88. Situation of its equator, 390... 
Maculiferous zones of, 393. Atmos- 
phere, 395. Its rose-colored clouds, 
395, Note 0. 395 b b. Relative 
illumination of centre and borders, 
386... 395. Temperature, 396. Ex- 
penditure of heat, 337. Action in 
producing winds, etc., 399. Specu- 
lation on cause of its heat, 400 and 
note, 905 a b. Eclipses, 420. Den- 
sity of, 449. Natural centre of plan- 
etary system, 462. Distance, how 
determined, 479. Its size illustrated, 
526. Action in producing tides, 751. 
Proper motion of, 854... 858 a b c, 
Note L. Absolute velocity of, in 
space, 858, Central, speculations 
on, 861. 

Sunsets, two witnessed in one day, 26. 

Superposition of small motions, 607. 

Survey, trigonometrical nature of, 274. 

Synodic revolution, 418. Of sun and 
moon, ib. 

System, solar, its motion in space, 
858... 



Tails of comets, spectra of, Note 0, 
395 d d. 

Tangential force and its effects, 618. 
Momentary action on perihelia, 673, 
Wholly influential on velocity, 660. 
Produces variations of axis, ib... 
Double the rate of advance of 
lunar apsides, 686. Of Neptune 
on Uranus and its effects, 774. 

Telegraphic signals, 259... 

Telescope, 54. Its application to as- 
tronomical instruments, 117. For 
viewing the sun. (See Helioscope.) 

Telescopie sights, invention of, 158 
note. 

Temperature of earth's surface at 
different seasons, 366. In South 
Africa and Australia, 369. Of the 
sun, 396. 

Temporary stars, 827.., 

Tethys, 548 note. 

Thales, eclipse of, 933 a, 926. Table. 

Theodolite, 192. Its use in surveying- 
276. 

Theory of instrumental errors, 141. 
Of gravitation, 430... Of nebulous 
subsidence and sidereal aggregation, 
872. 

Thomson (Prof. Sir W.), hi3 estimate 
of the sun's expenditure of heat, 
397 a, His theory of the source of 
do... 905 a. 

Tides, a system of forced oscillations, 
651. Explained, 750... Priming and 
lagging of, 753. Periodical inequali- 
ties of, 755. Instances of very high, 
756. 

Time, sidereal, 110, 327, 911. Local, 
129, 152, 252. Sidereal and solar, 
143. Mean and apparent sidereal, 
327. Measures angular motion, 149. 
How itself measured, 150... Very 



1XDEX 



925 



small intervals of, 150. Equinoc- 
tial, 25T, 925... Measures, units 
and reckoning of, 906... Required 
for light of stars to reach the earth, 

Titan, 54S note. 

Titius (Prof.), his law of planetary dis- 
tances, 505 note. 

Torsion balance used to measure den- 
sity of the earth, 776 i. Torsion 
balance to measure the variation of 
gravity, Note E. 

Trade winds, 239... 

Transit instrument, 159... 

Transits of stars, 152. Of planets 
across the sun, 467. Of Tenus, 
479... Mercury, 483. Of Jupiter's 
satellites across disk, 540. Of their 
shadows, 549. 

Transparency of space, supposed by 
Olbers imperfect, 798. 

Transversal disturbing force, and its 
effects, 615... 

Triangles ill-conditioned, 275. On an 
ellipsoid, 276. 

Trigonometrical survey, 274. 

Tropics, 93, 380. 

Twilight, 44. 

U 

Umbra in eclipses, 420. Of Jupiter, 
538. 

Umbriel, 551. 

TJranography, 111, 300. 

Uranographical corrections, 342... 
Problems, 127... 309... 

Uranometry, 118. 

Uranus, discovery of, 505. Heat re- 
ceived from sun by, 508. Physical 
description of, 523. Satellites of, 
551. Perturbations of, by Neptune, 
760... Old observations of, 760. 
Its periodic time, 776. Its action 
on the November meteors, 905 h h. 



Vanishing point of parallel lines, 116. 
Line of parallel planes, 117. 

Variable stars, 820 et seq. 

Variation of the moon explained, 705... 

Variations of elements, 653. Periodi- 
cal and secular, 655. Incident on 
the epoch, 731. 

Velocity, angular, of sun not uniform, 
350 Linear, of sun not uniform, 
351. Of planets, Mercury, Venus 
and Earth, 474. Of light, 545. 
Overestimated, its probable correc- 
tion, 357 a, Note F. Of shooting 
stars, 889, 904. Of Sirius, Note J, 
859 b b. 

Venus:, synodic revolution of, 472. 
Stationary points, 476. Velocity of, 
474. Phases, 477. Point of great- 
est biightness, 478. Transits of, 
479. Physical description and ap- 
pearance, 509. Inequality in earth's 
motion produced by, 726. In that 
of the moon, 743... 

Vernier, 165, 197. 

Vertical, prime, 102. Circles, 100. 

Vesta, discovery of, 505. 

Via Lactea. (See Galaxy.) 

Villarceau. (M. Vvon), his orbits of 
double stars, 843. 

Volcanoes, lunar, 430 a, Note H. 

W 

Watershed, 289. 

Watherson, his theory of the sun's 

heat, 905 a. 
Weight of bodies in .different latitudes, 

322. Of a body on the moon, 508. 

On the sun, 450. 
Wheatstone, his measurement of the 

velocity of electricity, 545. His 

method of stereographing the moon, 

437 a, Note I. 



926 



INDEX 



"Whipple, his photographs of the moon, 

437. 
Willow-leaved forms observed in the 

sun's photosphere, 387 a, Note G. 
Winds, trade, 240... 
Wiunecke (M.), his proposal for ob- 
serving the parallax of Mars, 357 a, 

Xote F. 
Witte (Madame), her models of the 

moon, 437. 
Wolf, liis period of the solar spots, 

394 a. 
Wollaston (Dr.), his estimate of the 

comparative light of the sun, moon 

and stars, 817... 



Year, sidereal, 305. Tropical, 383. 
Anomalistic, 384; and day incom- 
mensurable, 913. Leap, 914. Of 
confusion, 917, 932. Beginning of 
in England changed, 932. 



Zenith, 99. Sector, 192. 

Zodiac, 305. 

Zodiacal light, 897. 

Zones of climate and latitude, 38? 



Fiy4 




ri 9 .z. 




Fi.f.1 



Fif.3 




H-Adla.i-d..sc 



sffiorluyri of the Moons surface, from a model byjWNasrnyth. 



Fig-1 



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Pla.te B 









4 









scale creoooo muj:». 



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